Large Random Simplicial Complexes, II; the fundamental groups

In our recent work we described conditions under which a multi-parameter random simplicial complex is connected and simply connected. We showed that the Betti numbers of multi-parameter random simplicial complexes in one specific dimension dominate significantly the Betti numbers in all other dimensions. In this paper we focus mainly on the properties of fundamental groups of multi-parameter random simplicial complexes, which can be viewed as a new class of random groups. We describe thresholds for nontrivially and hyperbolicity (in the sense of Gromov) for these groups. Besides, we find domains in the multi-parameter space where these groups have 2-torsion. We also prove that these groups have never odd-prime torsion and their geometric and cohomological dimensions are either 0,1, 2 or infinity. Another result presented in this paper states that aspherical 2-dimensional subcomplexes of random complexes satisfy the Whitehead Conjecture, i.e. all their subcomplexes are also aspherical (with probability tending to one).


Introduction
This paper develops the multi-parameter model of random simplicial complexes initiated in [12] and [14].
One of the main motivations to study random simplicial complexes comes from the theory of large networks. Traditionally one models networks by graphs with nodes representing objects and edges representing connections between the objects [24]. However if we are interested not only in pairwise relations between the objects but also in relations between multiple objects we may use the high dimensional simplicial complexes instead of graphs as mathematical models of networks.
The mathematical theory of large random simplicial complexes is a new active research area, see [9] and [21] for surveys.
The multi-parameter model which we discuss here allows regimes controlled by a combination of probability parameters associated to various dimensions. This model includes the well-known Linial -Meshulam -Wallach model [22], [23] as an important special case; as another important special case it includes the random simplicial complexes arising as clique complexes of random Erdős-Rényi graphs, see [20], [11].
In the multi-parameter model one starts with a set of n vertices and retains each of them with probability p 0 ; on the next step one connects every pair of retained vertices by an edge with probability p 1 , and then fills in every triangle in the obtained random graph with probability p 2 , and so on. As the result we obtain a random simplicial complex depending on the set of probability parameters (p 0 , p 1 , . . . , p r ), 0 ≤ p i ≤ 1.
The topological and geometric properties of multi-parameter random simplicial complexes depend on the whole set of parameters and their thresholds can be understood as subsets of the space of multi-parameter space and not as single numbers as in all the previously studied models.
In our recent paper [14] we described the conditions under which a multiparameter random simplicial complex is connected and simply connected. In [13] we showed that the Betti numbers of multi-parameter random simplicial complexes in one specific dimension dominate significantly the Betti numbers in all other dimensions. In this paper we focus mainly on the properties of fundamental groups of multi-parameter random simplicial complexes, which can be viewed as a new class of random groups. We describe thresholds for nontrivially and hyperbolicity (in the sense of Gromov) for these groups. Besides, we find domains in the multi-parameter space where these groups have 2-torsion. We also prove that these groups have never odd-prime torsion and their geometric and cohomological dimensions are either 0, 1, 2 or ∞. Another result presented in this paper states that aspherical 2-dimensional subcomplexes of random complexes satisfy the Whitehead Conjecture, i.e. all their subcomplexes are also aspherical (with probability tending to one). Figure 1: Areas on the (α 1 , α 2 )-plane corresponding to various properties of the fundamental group: (a) light green -the group is trivial; (b) grey -the group has 2-torsion and is hyperbolic; (c) shaded black (including the horizontal interval (11/30, 2/5) shown in bold) -the group is nontrivial, hyperbolic, its geometric dimension is ≤ 2; (d) yellow -the group is trivial for any choice of the base point.
To make this paper less dependent on [14] we give now a brief description of the multi-parameter model. Fix an integer r ≥ 0 and a sequence p = (p 0 , p 1 , . . . , p r ) of real numbers satisfying 0 ≤ p i ≤ 1.
Denote q i = 1 − p i . We consider the probability space Ω r n consisting of all subcomplexes Y ⊂ ∆ n with dim Y ≤ r, where the symbol ∆ (r) n stands for the r-dimensional skeleton of ∆ n , which is defined as the union of all simplexes of dimension ≤ r. The probability function P r,p : Ω r n → R is given by the formula Here e i (Y ) denotes the number of i-dimensional external faces of Y , see [14] for the definition. We use the convention 0 0 = 1; in other words, if p i = 0 and f i (Y ) = 0 then the corresponding factor in (1) equals 1. Some results presented in the paper are illustrated by Figure 1. There for simplicity we assume that the probability parameters p i have the form where α i ≥ 0 are constant. We also assume that α 0 = 0 and hence various properties of the fundamental group are described by subsets of the (α 1 , α 2 )plane since they clearly depend only on p 1 and p 2 . Figure 1 shows domains corresponding to triviality and non-triviality, existence and non-existence of 2torsion as well as the domain showing when the geometric and cohomological dimension of the fundamental group are ≤ 2.
In area (a) the complex is connected and simply connected, as proven in [14]. In area (d) the random complex is disconnected and has no cycles and no 2-simplexes, i.e. it is a forest. Of course, all these properties are satisfied asymptotically almost surely (a.a.s.), i.e. the limit of the probability that the corresponding property holds tends to 1 as n → ∞. This research was supported by an EPSRC grant.

Subcomplexes of random complexes
Let S be a fixed simplicial complex of dimension ≤ r. Similarly to the random graph theory, one may wish to determine conditions when S is embeddable into a random simplicial complex Y ∈ Ω r n with respect to the probability measure P r,p , where p = (p 0 , p 1 , . . . , p r ). In a recent paper [12] we discussed the containment problem under a simplifying assumption that p i = n −αi where α i ≥ 0 are constant. In this paper we shall need the containment result without this assumption. Besides the general containment result stated as Theorem 1 we also treat in this section the relative containment problem, see Theorem 4. Theorem 1. Consider a random simplicial complex Y ∈ Ω r n with respect to the measure P r,p , where p = (p 0 , p 1 , . . . , p r ). Let S be a fixed finite simplicial complex of dimension ≤ r.
A. Suppose that Here T ⊂ S runs over all simplicial subcomplexes of S and f i (T ) denotes the number of i-dimensional faces in T . Then the probability that a random complex Y ∈ Ω r n contains a simplicial subcomplex isomorphic to S tends to zero as n → ∞.
B. Suppose that for any nonempty subcomplex T ⊂ S one has Then a random complex Y ∈ Ω r n contains a subcomplex isomorphic to S with probability tending to 1.
Proof. Let us start with the statement A. For T ⊂ S let X T : Ω r n → R denote the random variable counting the number of embeddings of T into a random simplicial complex Y ∈ Ω r n . One has Denoting by P r,p (Y ⊃ S) the probability that Y contains a subcomplex isomorphic to S, we have (for n fixed) By (2) for any ∈ (0, 1) there exists N such that for any n ≥ N there exists Then which shows that the RHS of (3) tends to zero when n → ∞ under our assumption (2). We now prove statement B. We follow the argument used in the proof of Lemma 3.4 from [12]. Let F 0 (S) denote the set of vertices of S and let J : F 0 (S) → [n] be an embedding. Consider the indicator random variable X J : Ω r n → {0, 1} taking the value 1 on a subcomplex Y ∈ Ω r n iff J extends to a simplicial embedding J : S → Y . Then X = J X J counts the number of copies of S in Y . We wish to show that X > 0 a.a.s.
We have By Chebyshev inequality, Our goal is to show that Var(X) E(X) 2 → 0. One has The product X J · X J is the indicator random variable for the containment of the simplicial complex J(S) ∪ J (S). Hence we have Denote by T the subcomplex T = J −1 (T ) ⊂ S. For a fixed subcomplex T ⊂ S the number of pairs of embeddings J, J : S → ∆ n such that J −1 (J(S) ∩ J (S)) = T is bounded above by where C T denotes the number of isomorphic copies of T in S. We obtain

It follows that
Var(X) → ∞ for all non-empty subcomplexes T ⊂ S. Example 2. Let C be a simplicial loop of length m ≥ 3. Then for any nonempty subcomplex T ⊂ C on a has f 1 (T ) ≤ f 0 (T ) and therefore From Theorem 1 we obtain that a random complex Y ∈ Ω r n contains C as a subcomplex assuming that np 0 p 1 → ∞.
Next we shall consider a relative version of Theorem 1. Suppose that S 1 ⊃ S 2 are two simplicial complexes; we want to describe conditions when a random simplicial complex Y ∈ Ω r n admits an embedding S 2 → Y which cannot be extended to an embedding S 1 → Y .
By the definition, an embedding of S 1 into Y is an injective simplicial map Theorem 3. Assume that for any nonempty subcomplex T ⊂ S 2 one has and besides, n f0(S1,S2) · r i=0 p fi(S1,S2) i → 0.
Then the number of embeddings of S 1 into a random simplicial complex Y ∈ Ω r n is smaller than the number of embeddings of S 2 into Y , a.a.s. In particular, under the assumptions (4), (5) there exists an embedding of S 2 → Y which does not extend to an embedding S 1 → Y , a.a.s.
Proof of Theorem 2. For i = 1, 2, let X i : Ω r n → Z be the random variable that counts the number of embeddings of S i into the random complex Y ∈ Ω r n . Our goal is to show that X 1 < X 2 , a.a.s. We have tends to zero because of our assumption (5).
There are two sequences x 1 = x 1 (n) and x 2 = x 2 (n) > 0 such that and The sequences satisfy the desired properties; indeed, we have For every n we have the inequality By the Markov inequality we have By Chebyschev's inequality Indeed, we have already seen in the proof of Theorem 1 that the assumption (4) implies that Var(X2) E(X2) 2 → 0 and hence Var(X2) x 2 2 tends to zero. Incorporating this information into (6) we obtain P r,n (X 1 < X 2 ) → 1.
be a finite family of finite simplicial complexes of dimension ≤ r containing a given simplicial complex S and satisfying the following conditions: (a) for every nonempty subcomplex T ⊂ S one has Then with probability tending to one, a random simplicial complex Y ∈ Ω r n admits an embedding of S which does not extend to an embedding S j → Y , for every j = 1, . . . , N .
The result follows automatically from Theorem 3. Examples when Theorem 4 can be applied to produce interesting results appear later, see §5 and §6.

Uniform Hyperbolicity
In this section we state a theorem about uniform hyperbolically of random simplicial complexes. This theorem plays a crucial role later in this paper.
First we recall the relevant definitions. Let X be a finite simplicial complex. For a simplicial loop in the 1-skeleton γ : S 1 → X (1) ⊂ X, we denote by |γ| the length of γ, i.e. the number of edges composing γ. If γ is null-homotopic, γ ∼ 1, we denote by A X (γ) the area of γ, i.e. the minimal number of 2-simplices in any simplicial filling V for γ. A simplicial filling (or a simplicial Van Kampen diagram) for a loop γ is defined as a pair of simplicial maps → X (where V is a contractible 2-dimensional complex) such that γ = b • i and the mapping cylinder of i is a disc with boundary S 1 × 0, see [2].
Next one defines the isoperimetric constant of X: Clearly I(X) = I(X (2) ), i.e. the isoperimetric constant I(X) depends only on the 2-skeleton X (2) . It is well-known that the positivity I(X) > 0 depends only on the fundamental group π 1 (X); in fact, one of the many equivalent definitions of hyperbolicity of discrete groups in the sense of M. Gromov [17] states that π 1 (X) is hyperbolic iff I(X) > 0. Clearly, the precise value of I(X) depends on the simplicial structure of X. Knowing I(X) (or a lower bound for it) is extremely useful as we shall demonstrate later in this paper. Below is the main result of this section: Theorem 5. Consider a random simplicial complex Y ∈ Ω r n with respect to the probability measure P r,p , where p = (p 0 , . . . , p r ), r ≥ 2. Assume that and for some > 0, Then there exists a constant c > 0 (depending only on ) such that a random complex Y ∈ Ω r n has the following property with probability tending to one: any subcomplex Y ⊂ Y satisfies In particular, any subcomplex Y ⊂ Y has a Gromov hyperbolic fundamental group, a.a.s..
Thus, under the assumption (11), all random complexes and their subcomplexes have isoperimetric constants bounded below by c > 0 with probability tending to 1 as n → ∞.
Assumption (10) guaranties that the number of vertices of Y tends to infinity, see [12], Lemma 2.5.
Using Theorem 5 we shall establish the following Corollary: [see Theorem 30] If additionally to the hypothesis of Theorem 5 one has as n → ∞ then a random complex Y ∈ Ω r n has non-trivial fundamental group, a.a.s.
To illustrate the importance of the assumption (13) note that the "alternative" assumption np 0 p 1 → 0 (14) implies (as is easy to show) that a random complex Y ∈ Ω r n is at most one dimensional and has no cycles, i.e. it is a forest, a.a.s. In particular, the fundamental group of Y is trivial with any base point, a.a.s. Note also that (14) implies that Y is disconnected, see Lemma 5.4 in [13].
The last part of Corollary 7 is proven in an earlier paper [14]. The Figure 2 depicts the domain where the fundamental group of the random complex is nontrivial. Here we assume that α 0 = α 3 = α 4 = · · · = 0.
Next we recall the local-to-global principle of Gromov which plays a crucial role in the proof of Theorem 5: Theorem 8. Let X be a finite 2-complex and let C > 0 be a constant such that any pure subcomplex S ⊂ X having at most (44) 3 · C −2 two-dimensional simplexes satisfies I(S) ≥ C. Then I(X) ≥ C · 44 −1 .
Now we give the following definition.
Definition 9. Let > 0 be a positive number. We shall say that a finite 2dimensional simplicial complex S is -admissible if the following system of linear inequalities admits a solution with α 1 ≥ 0, α 2 ≥ 0, where T ⊂ S runs over all non-empty subcomplexes. In other words, a 2-complex S is -admissible if there exist nonnegative real numbers α 1 and α 2 such that 3α 1 + 2α 2 > 1 + and for any non-empty subcomplex T ⊂ S one has α 1 f 1 (T ) + α 2 f 2 (T ) < f 0 (T ).
We shall say that a simplicial complex S is admissible if it is -admissible for some > 0.
The property of being -admissible is a combinatorial property of a 2-complex S which amounts to certain restrictions on the numbers of vertices, edges and faces for all subcomplexes of S. We may mention two special cases: Case A: A complex S is -admissible if for any subcomplex T ⊂ S one has This is equivalent to Definition 9 under an additional assumption that α 1 = 0. Complexes with µ 2 (S) > 1/2 were studies in §2 of [10]. Case B: A complex S is -admissible if for any subcomplex T ⊂ S one has This is equivalent to Definition 9 under an additional assumption that α 2 = 0. Complexes with µ 1 (S) > 1/3 were studies in §5 of [11].
Remark 11. A 2-complex S is said to be balanced if for any subcomplex T ⊂ S one has µ i (T ) ≥ µ i (S) for i = 1, 2; see §4 from [12]. Recall that µ i (T ) denotes the ratio f 0 (T )/f i (T ). A balanced 2-complex S is -admissible iff for some α 1 ≥ 0 and α 2 ≥ 0 satisfying 3α 1 + 2α 2 > 1 + . Note that any triangulation of a closed surface with non-negative Euler characteristic is balanced, see Theorem 4.4 from [12].
Example 12. Let S be a connected 2-complex homeomorphic to a closed surface with positive Euler characteristic, χ(S) > 0. Then S is -admissible assuming that Indeed, S is balanced (see above) and the well-known Euler type relations imply The inequality (16) can be rewritten in this case as and hence the system (15) is equivalent to We see that the existence of α 1 , α 2 follows from the inequality which is equivalent to (17). We obtain that any triangulated sphere or projective plane with sufficiently "small" number of vertices (as prescribed by (17)) isadmissible.
Besides, all triangulations of the 2-sphere and of the real projective plane are admissible.
Example 13. Any graph is -admissible. Indeed, since f 2 (T ) = 0, in the Definition 9 one may take α 1 very small and α 2 very large.
The importance of the notion of -admissibility stems from the following Lemma: Lemma 14. Assume that np 0 → ∞ and for some > 0, For a fixed constant C > 0, a random simplicial complex Y ∈ Ω r n with probability tending to one has the following property: any simplicial pure 2-dimensional subcomplex of Y with at most C 2-simplexes is -admissible. In other words, under the condition (18), consider the finite set F C, = {X} of isomorphism classes of pure 2-dimensional simplicial complexes X satisfying f 2 (X) ≤ C which are not -admissible. Then with probability tending to 1 a random complex Y ∈ Ω r n with respect to the multiparameter measure p = (p 0 , . . . , p r ) contains none of the complexes X ∈ F C, as a simplicial subcomplex.
Proof. We may write where in general α i = α i (n) is a function of n. By our assumption np 0 → ∞, we have α 0 (n) < 1 for all large n, i.e. for all n except finitely many. Our assumption (18) implies that where ω → ∞. This can be rewritten as Then β i (n) ≥ 0, and 3β 1 (n) + 2β 2 (n) > 1 + for any n. Let S be a simplicial complex with f 2 (S) ≤ C which is not -admissible. As follows from Definition 9, for any n there is a subcomplex T n ⊂ S such that which implies that Now we apply Theorem 1 to conclude that the probability that S is embeddable into Y tends to zero as n → ∞.
A crucial role in the proof of Theorem 5 plays the following theorem stating that all -admissible 2-dimensional complexes admit a universal lower bound on the value of their isoperimetric constant: Theorem 15. Given > 0 there exists a constant C > 0 such that for any -admissible finite simplicial pure 2-complex X one has I(X) ≥ C .
Proof of Theorem 5 using Theorem 8 and Theorem 15. Let C > 0 be the constant given by Theorem 15. Consider the set S of all isomorphism types of finite pure 2-complexes having at most C = 44 3 · C −2 two-dimensional simplexes. Clearly, the set S is finite. Let S ⊂ S denote the subset of complexes in S which are not -admissible. Under the assumptions of Theorem 5 a random complex Y ∈ Ω r n contains a complex from S as a subcomplex with probability tending to zero as n → ∞ as follows from Lemma 14.
All remaining complexes lying in S = S − S are -admissible. Theorem 15 states that any complex S ∈ S satisfies I(S) ≥ C . Now applying Theorem 8 we obtain that any subcomplex Y ⊂ Y satisfies I(Y ) ≥ C · 44 −1 = c with probability tending to 1 as n → ∞.
Theorem 15 will be proven in §10.

Topology of admissible 2-complexes
In this section we examine the topology of 2-complexes which are admissible in the sense of Definition 9. One of the central results proven here is Theorem 28 describing homotopy types of admissible 2-complexes. In §10 we shall continue the study of -admissible complexes and present a proof of Theorem 15.
Recall the definitions of the density invariants: We shall use the formulae see formula (8) in [11] and formula (2) in [10]. Here the sum is taken over the edges e of S and for an edge e the symbol deg e (the degree of e) denotes the number of 2-simplexes containing e.
Lemma 16. Let S be an -admissible 2-complex. Then for any subcomplex In particular, if S is admissible then for any subcomplex T ⊂ S one has either Proof. Suppose that for some T ⊂ S one has Then for any α 1 , α 2 ≥ 0 satisfying 3α 1 + 2α 2 ≥ 1 + one has which is equivalent to implying that S is not -admissible. The other statements follow from formulae (20).
Corollary 17. Let S be a simplicial 2-complex homeomorphic to a closed surface. If χ(S) ≤ 0 then S is not admissible.
Proof. Applying the previous Lemma 16 with T = S and observing that L(S) = 0 we have which shows that S is not admissible due to Lemma 16.
In Example 12 we showed that any closed surface S with χ(S) > 0 is admissible.
Recall that a 2-complex S is called closed if every edge e of S is contained in at least two 2-simplexes. Note that for a closed complex S one has L(S) ≤ 0. A 2-complex S is said to be pure if each edge e of S is contained in at least one 2-simplex.
Corollary 18. Any closed strongly connected 2-dimensional admissible simplicial complex S with b 2 (S) = 0 is either a triangulation of the real projective plane P 2 or the quotient Q 2 of a triangulation of P 2 obtained by identifying two adjacent edges.
Proof. By Lemma 16 we have that µ 1 (S) > 1/3. Lemma 5.1 of [11] implies that S is a triangulated projective plane P 2 or the quotient of a triangulation of P 2 obtained by identifying two adjacent edges.  2. One also obtains an admissible minimal cycle of type A by starting from a triangulation of S 2 and identifying two vertices. However, if one identifies more than two vertices the obtained minimal cycle is not admissible.
3. An admissible minimal cycle of type A is obtained from a triangulation of S 2 by identifying two adjacent edges.
4. An example of a minimal cycle of type B is given by the union Z = P 2 ∪ D 2 where P 2 is a triangulation of the real projective plane and D 2 is a triangulated disc such that P 2 ∩ D 2 = ∂D 2 is a non-contractible simple closed loop on P 2 . This minimal cycle is admissible if f 1 (∂D 2 ) ≤ 5.
5. Consider the union Z = P 2 ∪ P 2 of two real projective planes where the intersection P 2 ∩ P 2 is a loop non-contractible in each of the projective planes. Z is a minimal cycle of type B which is not admissible Remark 22. One can easily see that minimal cycles are closed, strongly connected simplicial complexes. Hence, for any minimal cycle Z one has L(Z) ≤ 0; besides, χ(Z) ≤ 2. Using Lemma 16 we find that every admissible minimal cycle Z must satisfy 3χ(Z) + L(Z) > 0 implying that Lemma 23. Any admissible minimal cycle Z of type A is homotopy equivalent to either S 2 or to S 2 ∨ S 1 . Moreover, for every 2-simplex σ ⊂ Z the boundary ∂σ is null-homotopic in Z − σ.
Proof. Suppose Z is an admissible minimal cycle of type A. Since Z is closed, one has L(Z) ≤ 0. Using the second part of Lemma 16, we obtain µ 1 (Z) > 1/3. Now we may apply Lemma 5.6 from [11]. Note that the statement of Lemma 5.6 from [11] requires that µ 1 (T ) > 1/3 for any subcomplex T ⊂ Z; however the proof presented in [11] uses only the assumption µ 1 (Z) > 1/3.
Next we establish the following simple fact about minimal cycles of type B which strengthens Remark 22.
and either b 2 (Z ) = 1 or b 2 (Z − Z ) = 1; either of these possibilities contradicts the minimality of Z. Hence the graph Γ must contain a cycle and in particular, the number of edges of Γ satisfies f 1 (Γ) ≥ 3. Each edge of Z is incident to at least two faces of Z and every edge of Γ ⊂ Z is incident to at least one face of Definition 25. Let Z be an admissible minimal cycle of type B. Any closed strongly connected proper subcomplex Z 0 ⊂ Z is called a core of Z.
Clearly, for any core Z 0 ⊂ Z one has b 2 (Z 0 ) = 0 (by minimality). Applying Corollary 18 we see that any core is homeomorphic either to P 2 or to P 2 with two adjacent edges identified.
Lemma 26. An admissible minimal cycle Z of type B has a unique core Z 0 ⊂ Z.
Proof. Let Z be an admissible minimal cycle of type B. Let us assume that Z has two distinct cores Z , Z ⊂ Z.
Consider the graphs By the arguments used in the proof of Lemma 24 we obtain that f 1 (Γ ) ≥ 3 and f 1 (Γ ) ≥ 3. The graphs Γ and Γ cannot be edge-disjoint since otherwise Z would have at least 6 edges of degree ≥ 3; the latter would give L(Z) ≤ −6 contradicting Lemma 24. This implies that f 1 (Z ∩ Z ) > 0 and therefore the union Z ∪ Z is strongly connected. We observe next that the union Z ∪ Z must coincide with Z. Indeed, if Z ∪ Z = Z then by minimality b 2 (Z ∪ Z ) = 0 and by Corollary 18 the union Z ∪ Z is either homeomorphic to P 2 or to the quotient Q 2 of P 2 where two adjacent edges are identified. But neither P 2 nor Q 2 admits a triangulation in which it is a union of two distinct closed proper subcomplexes. Here we use our assumption Z = Z . Now we may show that On the last step we used the observation The statement regarding Γ follows similarly.
We know that χ(Z ∪ Z ) = 2 and χ(Z ) = χ(Z ) = 1; therefore Suppose that the intersection S has no 2-faces. Then S is a connected graph and every edge of S has degree ≥ 4 in Z since it is incident to at least two faces of Z and two faces of Z . However, Thus we see that The Mayer-Vietoris exact sequences with rational coefficients for the covers Since H 1 (S; Q) = Q we obtain from (24), (25): In the beginning of the proof we have observed that every edge of Γ and of Γ has degree ≥ 3 in Z and that Γ and Γ cannot be edge-disjoint. Hence f 1 (Γ ∪ Γ ) ≤ 5 and therefore b 1 (Γ ∪ Γ ) ≤ 2 since any graph on at most five edges has at most 2 independent cycles. Let us show that the case b 1 (Γ ∪Γ ) = 2 is impossible. If b 1 (Γ ∪Γ ) = 2 then Γ ∪Γ is a square with one diagonal, it has 5 edges and each of the edges has degree 3 in Z (since L(Z) ≥ −5, see above). Moreover, in this case all other edges of Z have degree 2 (as again follows from L(Z) ≥ −5). Let v be one of the vertices of degree 3 in the graph Γ ∪ Γ .
Then v is incident to exactly three odd degree edges in Z (see similar argument in [11], proof of Lemma 5.7 on page 15). All edges of Z incident to v which do not belong to Γ ∪ Γ have degree 2 in Z. In particular the link Lk Z (v) of v in Z would be a graph with an odd number of odd degree vertices which is impossible. Therefore, we obtain that From (26) and (27) it follows that i.e. the graphs Γ and Γ possess a common cycle C. The isomorphisms (24) and (25) give We obtain that some integral multiple of the cycle C bounds a Z-chains in Z − S and in Z − S and the difference of these two chains will be a nontrivial 2-dimensional cycle c lying in Z − S = Z contradicting the minimality of Z. Here S denotes the union of interiors of all 2-simplexes of S. Note that the complexes Z − S, Z − S and S have no common 2-simplexes. This completes the proof.
Lemma 27. Any admissible minimal cycle Z of type B is homotopy equivalent to the sphere S 2 . Let Z 0 ⊂ Z denote the core of Z. Then for any 2-simplex σ ⊂ Z 0 the complement Z − Int(σ) is contractible.
Proof. Consider the complex Z − Int(σ) where σ is a 2-simplex, σ ⊂ Z 0 lying in the core. Starting from the complex Z − Int(σ) and collapsing subsequently faces across the free edges we shall arrive to a connected graph Γ, as follows from the uniqueness of the core (Lemma 26). Since χ(Z) = 2 (see Lemma 24) we find χ(Γ) = χ(Z − Int(σ)) = 1. Therefore Γ is a tree. Hence the complex Z − Int(σ) is contractible. This implies that Z is homotopy equivalent to the result of attaching a 2-cell to Z − Int(σ), hence Z S 2 .
Theorem 28. Any admissible 2-complex X is homotopy equivalent to a wedge of circles, spheres and projective planes.
Proof. We will act by induction on b 2 (X). If b 2 (X) = 0 and X is admissible then using Corollary 18 we see that each strongly connected component of X is homotopy equivalent to P 2 . Hence X is homotopy equivalent to a wedge of circles and projective planes. Assume now that the statement of the Theorem has been proven for all connected admissible 2-complexes X with b 2 (X) < k. Consider an admissible 2-complex X satisfying b 2 (X) = k > 0. Find a minimal cycle Z ⊂ X and observe that the homomorphism H 2 (Z; Z) = Z → H 2 (X; Z) induced by the inclusion is injective. Let σ ⊂ Z be a 2-simplex; if Z is of type B we shall assume that σ lies in the core Z 0 ⊂ Z. We shall use Lemmas 23 or 27 depending whether Z is of type A or B. The complex X = X − Int(σ) satisfies b 2 (X ) = k − 1 and is admissible and thus by induction X is homotopy equivalent to a wedge of circles, spheres and projective planes. Therefore, X is homotopy equivalent to X ∨ S 2 and hence X is homotopy equivalent to a wedge of circles, spheres and projective planes.
Corollary 29. The fundamental group of any admissible 2-complex X is the free product of several copies of Z and Z 2 . In particular, π 1 (X) is hyperbolic.
Corollary 29 is in some sense a weak version of Theorem 15 which will be proven later in §10.

Non-triviality of the fundamental groups of random simplicial complexes
Theorem 30. Let Y ∈ Ω r n be a random simplicial complex with respect to the probability measure P r,p where p = (p 0 , p 1 , . . . , p r ). Assume that and for some > 0, Then for some choice of the base point y 0 ∈ Y the fundamental group π 1 (Y, y 0 ) is nontrivial, a.a.s.
Remark 31. If the assumption (29) is replaced by the stronger assumption then Y ∈ Ω r n is connected, a.a.s. (see Corollary 7.2 from [13]). However under the assumption (29) a random complex might be disconnected, see §7 from [13]) and thus, the fundamental group π 1 (Y, y 0 ) might depend on the choice of the base point y 0 ∈ Y .
Proof of Theorem 30. Let C be the simplicial loop of length 4, i.e. the boundary of the square. Using Example 2, we note that our assumption (29) implies that a random complex Y ∈ Ω r n contains C as a subcomplex with probability tending to one.
We want to show that Y contains C as "an essential subcomplex", i.e. such that the inclusion C ⊂ Y induces a non-trivial homomorphism π 1 (C, y 0 ) → π 1 (Y, y 0 ). This would imply that π 1 (Y, y 0 ) = 1 for some choice of the base point. We shall use Theorem 4 to show the existence of essential embeddings C → Y .
Let c > 0 be the constant given by Theorem 5. The arguments of the proof which is presented below may seem formal, and to illustrate them we give in this paragraph a brief vague intuitive description. If an inclusion C ⊂ Y is not essential then C → Y can be extended to a simplicial map D → Y where D is a simplicial disc. Using the inequality I(Y ) ≥ c (which we may assume to be satisfied due to Theorem 15), we may assume that D has at most 4 · c −1 2-simplexes. Of course the map D → Y does not have to be injective; therefore the image of D → Y is a subcomplex S ⊂ Y with at most 4 · c −1 2-simplexes. There are finitely many isomorphism types of disc triangulations with at most 4 · c −1 2-simplexes and there are finitely many isomorphism types of their simplicial images S; those S's which are not -admissible appear with probability tending to 0 (due to Lemma 14). Below we formalise the properties of complexes S which may appear in this way and use Theorem 4 to show that there exists an embedding S → Y which cannot be extended to an embedding S → Y ; this embedding is clearly essential. Now we continue with the formal argument. Consider the set L of isomorphism types of pairs (S, C) where S is a finite 2-complex such that: (a) the inclusion C → S induces the trivial homomorphism of the funda-mental groups; (b) S is minimal in the sense that for any proper subcomplex C ⊂ S S the inclusion C → S induces a non-trivial homomorphism π 1 (C) → π 1 (S ); ( Note that for S ∈ L one has b 2 (S) = 0. Indeed, if b 2 (S) = 0 then S would contain a minimal cycle Z ⊂ S and this minimal cycle would be -admissible. By Lemma 23 and Lemma 27 the complex Z contains a 2-simplex σ such that the boundary ∂σ is null-homotopic in Z − Int(σ). Hence removing σ does not change the fundamental group and we obtain contradiction with the minimality property (b).
We obtain that χ(S) ≤ 1 for any S ∈ L . It is easy to see that S cannot have edges of degree zero (i.e. S must be pure) and any edge of S of degree 1 must lie in C (as follows from the minimality property (b)). Therefore L(S) ≤ f 1 (C) = 4.
We observe that for any S ∈ L one has Recall that the notation f i (S, C) stands for f i (S) − f i (C), the number of isimplexes lying in S − C. For any simplicial complex X of dimension ≤ 2 one has 3χ(X) + L(X) = 3f 0 (X) − f 1 (X), 2χ(X) + L(X) = 2f 0 (X) − f 2 (X).
Applying these equalities to S and to C and using We claim that our assumptions (29) and (30) imply that np 0 p 3 1 p 2 2 → 0 and p 1 p 2 2 → 0. Indeed, (29) implies that np 0 → ∞ and then (30) implies that np 0 p 3 1 p 2 2 → 0. It is easy to see that p 1 p 2 2 → 0 follows. Next we observe that the set L is finite and hence we may apply Theorem 4. Since all the assumptions of this theorem are satisfied we obtain that with probability tending to one, for Y ∈ Ω r n there exists an embedding C → Y which cannot be extended to an embedding S → Y for any S ∈ L .
Consider the set Ω n ⊂ Ω r n consisting of complexes Y ∈ Ω r n satisfying the following three conditions: (3) Y contains C as a subcomplex such that no complex S ∈ L can be embedded into Y extending the embedding C → Y .
By Theorem 15, Lemma 14, Theorem 4 and Example 2, the probability that Y belongs to Ω n tends to one as n → ∞. We show below that for every Y ∈ Ω n the fundamental group π 1 (Y, y 0 ) is non-trivial for some choice of the base point.
Let us explain that for Y ∈ Ω n the embedding C → Y is essential. For any embedding C ⊂ Y which is not essential there exists a simplicial disc D with at most 4 · c −1 2-simplexes and ∂D = C which can be simplicially mapped into Y extending the embedding C → Y (this follows from the inequality I(Y ) ≥ c −1 given by Theorem 5). The image of this disc is a subcomplex S ⊂ Y containing C which satisfies the properties (a), (c). Property (d) is satisfied because of Lemma 14. If the minimality property (b) is not satisfied, then instead of S we can consider an appropriate smaller subcomplex. Hence any non-essential embedding C → Y can be extended to an embedding S → Y for some S ∈ L . This completes the proof.
Remark 33. In the proof presented above we showed the existence of essential loops of length 4. The same argument applies for loops of any fixed length ≥ 4; it also applies to loops of length 3 under an additional assumption p 2 → 0. It is obvious that in the case when p 2 = 1 every loop of length 3 is the boundary of a 2-simplex in Y and hence our statement would be false for loops of length 3.

The existence of 2-torsion
Theorem 34. Let Y ∈ Ω r n be a random simplicial complex with respect to the probability measure P r,p where p = (p 0 , p 1 , . . . , p r ). Assume that and for some > 0, Then the fundamental group π 1 (Y ) has elements of order 2, a.a.s.
Remark 35. The assumptions of Theorem 34 imply that Y ∈ Ω r n connected, a.a.s. To show this one may use Corollary 7.2 from [14], which requires that ω → ∞ and ωp 1 − log(ω) → ∞ where ω = np 0 . By (35) we have n 2/5 p 2/5 Remark 36. In the special case when p i = n −αi with α i ≥ 0 constant, where i = 0, 1, . . . Theorem 34 states that the fundamental group of a random complex Y has a nontrivial element of order 2 assuming that This result was proven in [10] in the special case when α 1 = 0. See Figure 3.
Proof of Theorem 34. Consider the triangulation S 0 of the real projective plane P 2 shown in Figure 4; it has 6 vertices, 15 1-simplexes and 10 2-simplexes. It is the triangulation of P 2 having the smallest number of vertices. By Theorem 4.4 from [12] S 0 is balanced, which means that for any non-empty subcomplex Applying Theorem 1, B, se see that for any T ⊂ S 0 , T = ∅, Thus we see that under the assumption (35) the simplicial complex S 0 embeds into a random complex Y ∈ Ω r n , a.a.s.  We want to show that Y contains S 0 as "an essential subcomplex", i.e. such that the inclusion S 0 ⊂ Y induces a non-trivial homomorphism π 1 (S 0 ) = Z 2 → π 1 (Y ); this would imply that π 1 (Y ) has 2-torsion. We shall use Theorem 4 to show the existence of an essential embedding S 0 → Y . Our strategy will be similar to those used in the proof of Theorem 30.
We construct below a finite list L of 2-complexes S containing S 0 such that every non-essential embedding of S 0 into Y ∈ Ω n extends to an embedding S → Y , for some S ∈ L .
Consider the set L of isomorphism types of pairs (S, S 0 ) where S is a finite 2-complex containing S 0 satisfying the following conditions: (a) the inclusion S 0 → S induces the trivial homomorphism of the fundamental groups; (b) S is minimal in the sense that for any proper subcomplex S 0 ⊂ S S the inclusion S 0 → S induces an injective homomorphism π 1 (S 0 ) → π 1 (S ); (c) f 2 (S) ≤ 3 · c −1 + 10; (d) S is -admissible. Note that any S ∈ L is pure. Indeed, if S ⊂ S denotes the pure part of S then π 1 (S ) → π 1 (S) is injective and hence the inclusion S 0 ⊂ S induces a trivial homomorphism π 1 (S 0 ) → π 1 (S ); therefore the minimality property (b) implies S = S .
Consider the minimal cycles contained in a complex S ∈ L . If Z ⊂ S is a minimal cycle of type A then Z cannot be contained in S 0 and by Lemma 23 there is a 2-simplex σ ⊂ Z − S 0 with ∂σ null-homotopic in Z − Int(σ). Removing σ from S does not change the fundamental group and leads to a proper subcomplex S 0 ⊂ S ⊂ S contradicting the minimality property (b). Hence S does not contain minimal cycles of type A.
Let Z ⊂ S be a minimal cycle of type B where S ∈ L . Let Z 0 ⊂ Z be the core of Z. Recall that the core Z 0 is either homeomorphic to P 2 or to the quotient Q 2 of P 2 where two adjacent edges are identified. If Z 0 does not coincide with S 0 then by Lemma 27 we may find a 2-simplex σ ⊂ Z 0 − S 0 with ∂σ null-homotopic in Z − Int(σ). Removing σ from S does not change the fundamental group and leads to a subcomplex of S contradicting the minimality property (b). Hence Z 0 = S 0 .
This shows that S cannot have minimal cycles of type A and any minimal cycle of type B contained in S must have S 0 as its core.
Let Z ⊂ S be a minimal cycle of type B contained in S. Then S 0 ⊂ Zand by Lemma 27 Z is simply connected and therefore the inclusion S 0 ⊂ Z induces the trivial homomorphism on the fundamental groups. Hence by minimality (b) we have Z = S. This shows that b 2 (S) ≤ 1. In particular, (37) Next we show that L(S, S 0 ) = L(S) − L(S 0 ) ≤ −3. Let S = S − S 0 be the closure of the complement of S 0 in S. The intersection S 0 ∩ S is a graph Γ. If b 1 (Γ) = 0 then the inclusion S 0 → S is injective on the fundamental groups. Hence b 1 (Γ) ≥ 1 and thus f 1 (Γ) ≥ 3 implying that Note that for any simplicial complex X of dimension ≤ 2 one has 2χ(X) + L(X) = 2f 0 (X) − f 2 (X).
Using these equalities and the above relations we obtain We conclude that for any S ∈ L one has Now we see that Applying Theorem 4 we obtain that with probability tending to one, Y ∈ Ω r n admits an embedding of S 0 which does not extend to an embedding S → Y for every S ∈ L . Let c > 0 be the constant given by Theorem 5. Consider the set Ω n ⊂ Ω r n consisting of complexes Y ∈ Ω r n satisfying the following three conditions: (3) Y contains a copy of S 0 as a subcomplex such that there exists no complex S ∈ L for which the embedding S 0 → Y can be extended to an embedding S → Y .
By Theorem 5, Lemma 14, Theorem 1 and Theorem 4.4 from [12], the probability that Y belongs to Ω n tends to one as n → ∞.
Let us explain that for Y ∈ Ω n the embedding S 0 → Y is essential. For any embedding S 0 ⊂ Y which is not essential there exists a simplicial disc D with at most 3 · c −1 2-simplexes and ∂D = C where C ⊂ S 0 is the non-null-homotpic loop on S 0 of length 3. The disc D is simplicially mapped into Y extending the embedding C → Y (this follows from the inequality I(Y ) ≥ c −1 given by Theorem 5). The union of S 0 and the image of this disc is a subcomplex S ⊂ Y containing S 0 which satisfies the properties (a), (c). Property (d) is satisfied because of Lemma 14. If the minimality property (b) is not satisfied, then instead of S we can take an appropriate smaller subcomplex. Hence any nonessential embedding C → Y can be extended to an embedding S → Y for some S ∈ L .
This completes the proof.
Note that the assumption p 2 → 0 was essentially used in the proof since in the product (40) the exponent f 0 (S, S 0 ) may happen to be 0. We believe that Theorem 34 will remain true under a weaker assumption p 2 < c < 1 where c is a constant.
Next we state another result assuming that p 2 = 1, which is a generalisation of Theorem 7.2 from [11].
Theorem 37. Let Y ∈ Ω r n be a random simplicial complex with respect to the probability measure P r,p where p = (p 0 , p 1 , . . . , p r ). Assume that and for some > 0, Then the fundamental group π 1 (Y ) has nontrivial elements of order two, a.a.s.
Remark 38. In the special case when . . Theorem 37 states that the fundamental group of a random complex Y has a nontrivial element of order 2 assuming that This result was proven in [11] in the special case when α 2 = 0.
Remark 39. The main distinctions between Theorems 34 and 37 are the assumptions regarding the behaviour of p 2 . The 2-torsion in the fundamental group is generated by essential embeddings of the real projective plane P 2 . In the case of Theorem 34 we are dealing with the embeddings of the minimal triangulation S 0 of P 2 . However, in the case when p 2 = 1 the triangular essential loop of S 0 bounds a triangle in Y . This explains that in the case p 2 = 1 one has consider clean triangulations of P 2 . Recall that a triangulation of a 2-complex is called clean if for any clique of 3 vertices {v 0 , v 1 , v 2 } the complex contains also the 2-simplex (v 0 , v 1 , v 2 ). We shall use the following fact: any clean triangulation of the projective plane P 2 contains at least 11 vertices and 30 edges, see [18]. The minimal clean triangulation is shown in Figure 5; the antipodal points of the circle must be identified.
Proof of Theorem 37. The proof is very similar to the proof of Theorem 34; we shall indicate the main steps and emphasise the main distinctions.
Let S 0 be the minimal clean triangulation of the real projective plane shown on Figure 5. It is balanced (by Theorem 4.4 from [12]) and as in the proof of Theorem 34 we find that for np 0 p 30/11 1 p 20/11 2 → ∞ the complex S 0 simplicially Figure 5: The minimal clean triangulation of P 2 , according to [18]. embeds into Y , a.a.s. Since we assume that p 2 = 1, this conditions coincides with (42).
As above, we want to show that Y contains S 0 as "an essential subcomplex".
Consider the set L of isomorphism types of pairs (S, S 0 ) where S is a finite 2-complex containing S 0 satisfying the following conditions: (a) the inclusion S 0 → S induces the trivial homomorphism of the fundamental groups; (b) S is minimal in the sense that for any proper subcomplex S 0 ⊂ S S the inclusion S 0 → S induces an injective homomorphism π 1 (S 0 ) → π 1 (S ); (c) f 2 (S) ≤ 4 · c −1 + 20; (d) S is -admissible. As in the proof of Theorem one obtains that each S ∈ L is pure and and we see that Note that p 1 → 0 as follows from (43). The rest of the proof is identical to the proof of Theorem 34. This completes the proof.

Higher torsion
In this section we show that random simplicial complexes have no odd torsion for a large range of probability parameters.
Theorem 40. Let m ≥ 3 be a fixed odd prime. Consider a random simplicial complex Y ∈ Ω r n , r ≥ 2, with respect to the probability measure P r,p where p = (p 0 , p 1 , . . . , p r ). Assume that for some > 0 one has Then a random complex Y ∈ Ω r n with probability tending to 1 has the following property: the fundamental group of any connected subcomplex Y ⊂ Y has no m-torsion.
Remark 41. In the special case when p i = n −αi with α i ≥ 0 being constant, where i = 0, 1, . . . Theorem 40 states that for any odd prime m ≥ 3 the fundamental group of a random complex Y has no nontrivial elements of order m assuming that This result was proven in [10] in the special case when α 1 = 0 and in [11] in the special case α 2 = 0.
The proof of Theorem 40 given below uses an auxiliary material which we now describe.
Let f m : S 1 → S 1 denote the canonical degree m map, f m (z) = z m where z ∈ S 1 ; we think of S 1 as being the unit circle on the complex plane. Any simplicial complex Σ homeomorphic M (Z m , 1) = S 1 ∪ fm e 2 is called a Moore surface.
Everywhere in this section we shall assume that m ≥ 3 is a fixed odd prime. Then any Moore surface Σ has a well defined circle C ⊂ Σ (called the singular circle) which is the union of all edges of degree m; all other edges of Σ have degree 2. Clearly, the homotopy class of the singular circle generates the fundamental group π 1 (Σ) Z m .
Define an integer N m (Y ) ≥ 0 associated to any connected simplicial complex Y ∈ Ω r n . If π 1 (Y ) has no m-torsion we set N m (Y ) = 0. If π 1 (Y ) has elements of order m we consider homotopically nontrivial simplicial maps γ : C r → Y , where C r is the simplicial circle with r edges, such that (a) γ m is null-homotopic (as a free loop in Y ); (b) r is minimal: for r < r any simplicial loop γ : C r → Y satisfying (a) is homotopically trivial.
Any such simplicial map γ : C r → Y can be extended to a simplicial map f : Σ → Y of a Moore surface Σ such that the singular circle C of Σ is isomorphic to C r and f |C = γ. We shall say that a simplicial map f : Σ → Y is m-minimal if it satisfies (a), (b) and the number of 2-simplexes in Σ is the smallest possible. Clearly, any m-minimal map Σ → Y induces an injective homomorphism π 1 (Σ) Z m → π 1 (Y ). We denote by Using this lemma we may obtain a global upper bound on the numbers N m (Y ) for random complexes: Theorem 43. Assume that the probability multi-parameter p = (p 0 , p 1 , . . . , p r ), where r ≥ 2, satisfies (np 0 ) 1+ p 3 1 p 2 2 → 0 with > 0 is fixed. Let m ≥ 3 be an odd prime. Then there exists a constant C > 0 such that a random complex Y ∈ Ω r n with probability tending to 1 has the following property: for any subcomplex Y ⊂ Y one has Proof. We know from Theorem 5 that, with probability tending to 1, a random complex Y has the following property: for any subcomplex Y ⊂ Y one has I(Y ) ≥ c > 0 where c > 0 is the constant given by Theorem 5. Then, setting two-dimensional simplexes. Let X m denote the set of isomorphism types of images of all surjective simplicial maps Σ → X inducing injective homomorphisms π 1 (Σ) = Z m → π 1 (X), where Σ ∈ S m . The set X m is also finite.
From Theorem 43 we obtain that, with probability tending to one, for any subcomplex Y ⊂ Y , either π 1 (Y ) has no m-torsion, or there exists an mminimal map f : Σ → Y where Σ is a Moore surface having at most 6m c 2-simplexes of dimension 2; in the second case the image X = f (Σ) is a subcomplex of Y and f : Σ → X induces a monomorphism π 1 (Σ) → π 1 (X), i.e. X ∈ X m . Denote by X m ⊂ X m the set of complexes X ∈ X m such that their 2-dimensional pure parts are -admissible. We may apply Lemma 14 (using the upper bound f 2 (X) ≤ C = 6m c ) to conclude that the pure part of the image X = f (Σ) of any m-minimal map f : Σ → Y belongs to X m . However, by Theorem 28 the fundamental group of any X ∈ X m is a free product of several copies of Z and Z 2 and hence it has no m-torsion. This leads to a contradiction which shows that the fundamental group of any subcomplex Y ⊂ Y does not have m-torsion, a.a.s.

Asphericity and the Whitehead Conjecture
Recall that a connected simplicial complex Y is said to be aspherical if π k (Y ) = 0 for all k ≥ 2. A 2-dimensional connected simplicial complex Y is aspherical if and only if π 2 (Y ) = 0. The well-known Whitehead Conjecture states that any subcomplex of an aspherical 2-complex is also aspherical, see [1,5,6,26]. At the time of writing the Whitehead Conjecture is still open.
In this section we show that for random simplicial 2-complexes, with probability tending to one, the asphericity of a subcomplex of a random complex is equivalent to the absence of "small bubbles". It follows that in the random setting any subcomplex of an aspherical 2-complex is also aspherical, supporting a probabilistic analogue of the Whitehead Conjecture.
Theorem 44. Consider a random simplicial complex Y ∈ Ω r n , r ≥ 2, with respect to the probability measure P r,p where p = (p 0 , p 1 , . . . , p r ). Assume that for some > 0 one has (np 0 ) 1+ p 3 1 p 2 2 → 0. Let c be the constant given by Theorem 5. Then, a random complex Y ∈ Ω r n has the following property with probability tending to 1 as n → ∞: any subcomplex Y ⊂ Y (2) is aspherical if and only if every pure subcomplex S ⊂ Y satisfying is collapsible to a graph. In particular, under the above assumptions, with probability tending to one, any aspherical subcomplex of Y (2) satisfies the Whitehead Conjecture.
The proof given below in this section will use the following auxiliary material. Let Y be a simplicial complex with π 2 (Y ) = 0. As in [10] and [11], we define a numerical invariant M (Y ) ∈ Z, M (Y ) ≥ 4, as the minimal number of faces in a 2-complex Σ homeomorphic to the sphere S 2 such that there exists a homotopically nontrivial simplicial map Σ → Y .
Lemma 45 (See Corollary 5.4 in [10]). Let Y be a 2-complex with I(Y ) ≥ c > 0. Then Combining this lemma with Theorem 5 we obtain: Lemma 46. Assume that for some > 0. Then there exists a constant C > 0 such that a random complex Y ∈ Ω r n has the following property with probability tending to one: for any subcomplex Y ⊂ Y (2) one has Hence, if we want to find homotopically nontrivial simplicial maps from S 2 to a random complex it is sufficient to consider triangulations of S 2 having at most C 2-simplexes.
The following lemma will be used in the proof of Theorem 44.
Lemma 47. Let S be a connected admissible 2-complex (see Definition 9). Then S is aspherical if and only if S is simplicially collapsible to a graph.
Proof. Let S be a connected admissible 2-complex satisfying π 2 (S) = 0. Performing all possible simplicial collapses of 2-simplexes we may find a closed pure subcomplex S ⊂ S without free edges. We need to show that S is 1-dimensional. We shall assume below that dim S = 2 and arrive to a contradiction.
If b 2 (S ) ≥ 1 then S contains an admissible minimal cycle Z ⊂ S as a subcomplex. Using Lemmas 23 and 27 (depending on whether Z is of type A or B) we find a 2-simplex σ ⊂ Z such that ∂σ is null-homotopic in Z − Int(σ). Then S is homotopy equivalent to the wedge S − Int(σ) ∨ S 2 contradicting the assumption π 2 (S ) = 0.
Hence we must assume that b 2 (S ) = 0. Using Corollary 18 we see that every strongly connected component of S is homeomorphic to either P 2 or to the quotient Q 2 of P 2 with two adjacent edges identified. In both cases we may apply a theorem of Cockcroft [6] (see also Adams [1]) which claims that π 2 (S ) = 0, in contradiction with our hypothesis.
Lemma 48. Let S be a connected admissible 2-complex. If S is not aspherical then S contains a subcomplex which is homotopy equivalent to one of S 2 , S 2 ∨S 1 or P 2 .
Proof. Let S be a connected admissible 2-complex; without loss of generality we may assume that S is closed and pure. If b 2 (S) = 0 then S contains an admissible minimal cycle Z ⊂ S and Z is homotopy equivalent to either S 2 , S 2 ∨ S 1 (in the case of type A, see Lemma 23), or to P 2 (in the case of type B, see Lemma 27). In the case b 2 (S) = 0 we invoke Corollary 18.
Proof of Theorem 44. Consider the set S of isomorphism types of all pure 2complexes S satisfying f 2 (S) ≤ C. We may represent S as the disjoint union S = S ∪ S where the complexes S ∈ S are -admissible and the complexes S ∈ S are not.
Let Ω n ⊂ Ω r n be the set of complexes Y ∈ Ω r n such that (a) no S ∈ S can be embedded into Y and (b) each Y ∈ Ω n satisfies the conclusion of Lemma 46. By Lemmas 14 and 46 we know that P r,p (Ω n ) → 1 as n → ∞.
Suppose that Y ∈ Ω n and let Y ⊂ Y (2) be an aspherical subcomplex. For any pure subcomplex S ⊂ Y with f 2 (S) ≤ C we know that S ∈ S and by Lemma 48 either S is aspherical or it contains a subcomplex homotopy equivalent to either S 2 or P 2 . Both these possibilities would imply π 2 (Y ) = 0 (the case of S 2 is obvious and the case of P 2 follows from the work of Cockcroft [6], see also [1]. Thus we see that in Y ⊂ Y is aspherical then any subcomplex S ⊂ Y with f 2 (S) < C is also aspherical; the latter due to Lemma 47 is equivalent for S to be collapsible to a graph.
We now prove the inverse implication by assuming that Y ⊂ Y is nonaspherical and Y ∈ Ω n . By Lemma 46 there exists a 2-complex Σ homeomorphic to a 2-sphere with f 2 (Σ) ≤ C and a homotopically nontrivial simplicial map φ : Σ → Y . We denote S = φ(Σ) and thus we have f 2 (S) ≤ f 2 (Σ) ≤ C and π 2 (S) = 0. Hence we conclude that if Y ⊂ Y is not aspherical then there exists a subcomplex S ⊂ Y , S ∈ S , and f 2 (S) ≤ C and π 2 (S) = 0.
This completes the proof.
9 Geometric and cohomological dimension of the fundamental group of a random simplicial complex Theorem 49. Consider a random simplicial complex Y ∈ Ω r n with respect to the multi-parameter probability measure P r,p , where p = (p 0 , p 1 , . . . , p r ), r ≥ 2. Assume that Then for any choice of the base point y 0 ∈ Y the fundamental group π 1 (Y, y 0 ) has geometric dimension at most 2, a.a.s. In particular, the group π 1 (Y, y 0 ) has cohomological dimension at most 2 and is torsion free, a.a.s.
Proof. We will show that with probability tending to one a random complex Y ∈ Ω r n contains a 2-dimension subcomplex Y ⊂ Y such that (2) π 1 (Y , y 0 ) π 1 (Y, y 0 ) for any vertex y 0 ∈ Y ; (3) any connected component of Y is aspherical.
This would clearly imply the statement of Theorem 49. Note that our assumption (50) implies the condition (11) for any < 1/5. Hence we may apply Theorem 5 and Theorem 44. To be specific we may set = 1/10 and denote by c > 0 the constant given by Theorem 5.
Denote by Ω n ⊂ Ω r n the set of complexes Y ∈ Ω r n satisfying the following conditions: (a) For Y ∈ Ω n any 2-dimensional pure subcomplex S ⊂ Y with f 2 (S) ≤ C is admissible. Here C denotes 16 2 c 2 .
(b) For Y ∈ Ω n a subcomplex Y ⊂ Y (2) is aspherical if and only if every subcomplex S ⊂ Y satisfying f 2 (S) < C is collapsible to a graph.
We know that P r,p (Ω n ) → 1 due to Lemma 14 and Theorem 44; to explain that the property (c) can be achieved we observe that by Corollary 18 any such S is either a triangulation of P 2 or the quotient Q 2 of P 2 obtained by identifying two adjacent edges in certain triangulation. Assuming that S is a triangulation of P 2 we have f 0 (S) − f 1 (S) + f 2 (S) = 1, 3f 2 (S) = 2f 1 (S) and f 0 (S) ≥ 6 implying that f 1 (S) ≥ 15 and f 2 (S) ≥ 10 and also The proof is similar to the proof of Theorem 49. The only difference is that in the case when p 2 = 1 we have a different requirement (51) on the absence of embeddings of clean triangulations of the real projective plane P 2 into a random 2-complex. Remark 51. In the special case when p i = n −αi with α i ≥ 0 constants, where i = 0, 1, . . . Theorem 49 states that the fundamental group of a random complex Y has geometric dimension at most 2 assuming that either or α 0 + 30 11 α 1 > 1, and α 2 = 0. See Figure 6; the corresponding set is represented by the shaded area together with an open interval on the α 1 axis. These results were proven in [10] in the special case when α 1 = 0 and in [11] in the case when α 2 = 0.

Proof of Theorem 15
Definition 52. (See [10]) We say that a finite 2-complex X is tight if for any proper subcomplex X ⊂ X, X = X, one has I(X ) > I(X).

Clearly, one has
By (52) it is obvious that it is enough to prove Theorem 15 under the additional assumption that X is tight.
Remark 53. Suppose that X is pure and tight and suppose that γ : S 1 → X is a simplicial loop with the ratio |γ|·A X (γ) −1 less than the minimum of the numbers I(X ) where X ⊂ X is a proper subcomplex. Let b : D 2 → X be a minimal spanning disc for γ; then b(D 2 ) = X, i.e. b is surjective. Indeed, if the image of b does not contain a 2-simplex σ then removing it we obtain a subcomplex X ⊂ X with A X (γ) = A X (γ) and hence I(X ) ≤ I(X) ≤ |γ| · A X (γ) −1 contradicting the assumption on γ.
Lemma 54. If X is a admissible tight complex then b 2 (X) = 0.
Proof. Assume that b 2 (X) = 0. Then there exists a admissible minimal cycle Z ⊂ X. Hence, by Lemmas 23 and 27 we may find a 2-simplex σ ⊂ Z ⊂ X such that ∂σ is null-homotopic in Z − σ ⊂ X − σ = X . Note that X (1) = X (1) and a simplicial curve γ : S 1 → X is null-homotopic in X if and only if it is null-homotopic in X. Besides, A X (γ) ≤ A X (γ) and hence which implies that I(X) ≥ I(X ). We obtained a contradiction since X is tight.
Lemma 55. Given > 0 there exists a constant C > 0 such that for every finite pure tight connected -admissible complex X satisfying L(X) ≤ 0 one has I(X) ≥ C .
This lemma is similar to Theorem 15 but it has an additional assumption that L(X) ≤ 0. The assumption L(X) ≤ 0 can be replaced, without altering the proof, by any assumption of the type L(X) ≤ 1000, i.e. by any specific upper bound.
Proof of Lemma 55. We show that the number of isomorphism types of complexes X satisfying the conditions of the lemma is finite; hence the proof follows by setting C = min I(X) and using Theorem 28 which gives I(X) > 0 (since π 1 (X) is hyperbolic) and hence C > 0.
By Lemma 16 we obtain The inequality where f 1 (X) denotes the number of 1-simplexes in X. By Lemma 54 we have χ(X) = 1 − b 1 (X) ≤ 1 and using the assumption L(X) ≤ 0 we obtain f 1 (X) ≤ −1 . This implies the finiteness of the set of possible isomorphism types of X and the result follows. The case µ 2 (X) ≥ (1 + )/2 is analogous.
We will use a relative isoperimetric constant I(X, X ) ∈ R for a pair consisting of a finite 2-complex X and its subcomplex X ⊂ X; it is defined as the infimum of all ratios |γ| · A X (γ) −1 where γ : S 1 → X runs over simplicial loops in X which are null-homotopic in X. Clearly, I(X, X ) ≥ I(X) and I(X, X ) = I(X) if X = X. Below is a useful strengthening of Lemma 55.
Lemma 56. Given > 0, let C > 0 be the constant given by Lemma 55. Then for any finite pure, tight, -admissible and connected 2-complex and for a connected subcomplex X ⊂ X satisfying L(X ) ≤ 0 one has I(X, X ) ≥ C .
Proof. We show below that under the assumptions on X, X one has where Y runs over all subcomplexes X ⊂ Y ⊂ X satisfying L(Y ) ≤ 0. Clearly, any such Y is -admissible. By Lemma 54 we have that b 2 (X) = 0 which implies that b 2 (Y ) = 0. Besides, without loss of generality we may assume that Y is connected. The arguments of the proof of Lemma 55 now apply (i.e. Y may have finitely many isomorphism types, each having a hyperbolic fundamental group) and it follows that min Y I(Y ) ≥ C where C > 0 is a constant that only depends on . Hence if (53) holds we have I(X, X ) ≥ min Y I(Y ) ≥ C and the result follows. Suppose that inequality (53) is false, i.e. I(X, X ) < min Y I(Y ), and consider a simplicial loop γ : S 1 → X satisfying γ ∼ 1 in X and |γ| · A X (γ) −1 < min Y I(Y ). Let ψ : D 2 → X be a simplicial spanning disc of minimal area. It follows from the arguments of Ronan [25], that ψ is non-degenerate in the following sense: for any 2-simplex σ of D 2 the image ψ(σ) is a 2-simplex and for two distinct 2-simplexes σ 1 , σ 2 of D 2 with ψ(σ 1 ) = ψ(σ 2 ) the intersection σ 1 ∩σ 2 is either ∅ or a vertex of D 2 . In other words, we exclude foldings, i.e. situations such that ψ(σ 1 ) = ψ(σ 2 ) and σ 1 ∩ σ 2 is an edge. Consider Z = X ∪ ψ(D 2 ). Note that L(Z) ≤ 0. Indeed, since where e runs over the edges of Z, we see that for e ⊂ X , deg X (e) ≤ deg Z (e) and for a newly created edge e ⊂ ψ(D 2 ), clearly deg Z (e) ≥ 2. Hence, L(Z) ≤ L(X ) ≤ 0. On the other hand, A X (γ) = A Z (γ) and hence I(Z) ≤ |γ| · A X (γ) −1 < min Y I(Y ), a contradiction.
The main idea of the proof of Theorem 15 in the general case is to find a planar complex (a "singular surface") Σ, with one boundary component ∂ + Σ being the initial loop and such that "the rest of the boundary" ∂ − Σ is a "product of negative loops" (i.e. loops satisfying Lemma 56). The essential part of the proof is in estimating the area (the number of 2-simplexes) of such Σ.
Proof of Theorem 15. Consider a connected tight pure -admissible 2-complex X and a simplicial prime loop γ : S 1 → X such that the ratio |γ| · A X (γ) −1 is less than the minimum of the numbers I(X ) for all proper subcomplexes X ⊂ X. Consider a minimal spanning disc b : D 2 → X for γ = b| ∂D 2 ; here D 2 is a triangulated disc and b is a simplicial map. As we showed in Remark 53, the map b is surjective. As explained in the proof of Lemma 56, due to arguments of Ronan [25], we may assume that b has no foldings.
For any integer i ≥ 1 we denote by X i ⊂ X the pure subcomplex generated by all 2-simplexes σ of X such that the preimage b −1 (σ) ⊂ D 2 contains ≥ i twodimensional simplexes. One has X = X 1 ⊃ X 2 ⊃ X 3 ⊃ . . . . Each X i may have several connected components and we will denote by Λ the set labelling all the connected components of the disjoint union i≥1 X i . For λ ∈ Λ the symbol X λ will denote the corresponding connected component of i≥1 X i and the symbol i = i(λ) ∈ {1, 2, . . . } will denote the index i ≥ 1 such that X λ is a connected component of X i , viewed as a subset of i≥1 X i . We endow Λ with the following partial order: λ 1 ≤ λ 2 iff X λ1 ⊃ X λ2 (where X λ1 and X λ2 are viewed as subsets of X) and i(λ 1 ) ≤ i(λ 2 ).
Finally we consider the following subcomplex of the disk D 2 : and we shall denote by Σ the connected component of Σ containing the boundary circle ∂D 2 .
Recall that for a 2-complex X the symbol f 2 (X) denotes the number of 2-simplexes in X. We have and Formula (55) follows from the observation that any 2-simplex of X = b(D 2 ) contributes to the RHS of (55) as many units as its multiplicity (the number of its preimages under b). Formula (56) follows from (55) and from the fact that for a 2-simplex σ of Σ the image b(σ) lies always in the complexes X λ with L(X λ ) > 0.
Lemma 57. One has the following inequality See [10], Lemma 6.8 for the proof. Now we continue with the proof of Theorem 15. Consider a tight -admissible pure 2-complex X and a simplicial loop γ : S 1 → X as above. We will use the notation introduced earlier. The complex Σ is a connected subcomplex of the disk D 2 ; it contains the boundary circle ∂D 2 which we will denote also by ∂ + Σ. The closure of the complement of Σ, is a pure 2-complex. Let N = ∪ j∈J N j be the strongly connected components of N . Each N j is PL-homeomorphic to a disc and we define ∂ − Σ = ∪ j∈J ∂N j , the union of the circles ∂N j which are the boundaries of the strongly connected components of N . It may happen that ∂ + Σ and ∂ − Σ have nonempty intersection. Also, the circles forming ∂ − Σ may not be disjoint.
We claim that for any j ∈ J there exists λ ∈ Λ − such that b(∂N j ) ⊂ X λ . Indeed, let λ 1 , . . . , λ r ∈ Λ − be the minimal elements of Λ − with respect to the partial order introduced earlier. The complexes X λ1 , . . . , X λr are connected and pairwise disjoint and for any λ ∈ Λ − the complex X λ is a subcomplex of one of the sets X λi , where i = 1, . . . , i. From our definition (54) it follows that the image of the circle b(∂N j ) is contained in the union ∪ r i=1 X λi but since b(∂N j ) is connected it must lie in one of the sets X λi .
We may apply Lemma 56 to each of the circles ∂N j . We obtain that each of the circles ∂N j admits a spanning discs of area ≤ K |∂N j |, where K = C −1 is the inverse of the constant given by Lemma 56. Using the minimality of the disc D 2 we obtain that the circles ∂N bound in D 2 several discs with the total area A ≤ K · |∂ − Σ|.
This completes the proof of Theorem 15.