Fundamental groups of random clique complexes

We study fundamental groups of clique complexes associated to random graphs. We establish thresholds for their cohomological and geometric dimension and torsion. We also show that in certain regime any aspherical subcomplex of a random clique complex satisfies the Whitehead conjecture, i.e. all irs subcomplexes are also aspherical.


Introduction
A clique in a graph Γ is a set of vertices of Γ such that any two of them are connected by an edge.The family of cliques of Γ forms a simplicial complex XΓ with the vertex set V (XΓ) equal the vertex set V (Γ) of Γ.The complex XΓ is called the clique complex (or the flag complex) of Γ.Clearly, the 1-skeleton of XΓ is the graph Γ itself.
In this paper we consider the clique complexes XΓ of random Erdős -Rényi graphs Γ ∈ G(n, p).Recall that G(n, p) is the probability space of all subgraphs Γ of the complete graph on n vertices satisfying V (Γ) = {1, . . ., n}, where the probability of a graph Γ equals )−e(Γ) .
Here p ∈ (0, 1) is a probability parameter, which in general is a function of n, and e(Γ) denotes the number of edges in Γ.The complex XΓ, where Γ ∈ G(n, p), is a random simplicial complex.One is interested in topological properties of XΓ which are satisfied with high probability when the number of vertices n tends to infinity.Topology of clique complexes of random graphs were studied by M. Kahle et al. in a series of papers [20], [21], [22], [23].A recent survey is given in [24].The following result is stated in a simplified form.
In this paper we are interested in the properties of the fundamental group of a random clique complex and therefore (see above) we shall restrict our attention to the regime α < −1/3 where p = n α .In a recent preprint [5] E. Babson proved that for ǫ > 0 and n ǫ−1/2 < p < n n−ǫ−1/3 the fundamental group π1(XΓ) is nontrivial and is hyperbolic in the sense of Gromov [16].
In this paper we use the notation f ≪ g to indicate that f /g → 0 as n → ∞.
The main results of this paper are as follows: then, with probability tending to 1 as n → ∞, the clique complex XΓ is simplicially collapsible to a graph.
In particular under the above assumption the fundamental group π1(XΓ, x0) of a random clique complex XΓ, where Γ ∈ G(n, p), is free, for any choice of the base point x0 ∈ XΓ, a.a.s.Moreover,each connected component of the 2-skeleton X (2) Γ is homotopy equivalent to a wedge of circles and 2-spheres, a.a.s.
Note that in the range (1) the dimension of XΓ is ≤ 3 and the 2-skeleton X Γ contains the tetrahedron and its subdivision having 5 vertices, a.a.s.Hence, the 2-skeleton X (2) Γ is not collapsible to a graph.Note also that for p ≫ n −1/2 the fundamental group π1(XΓ) ceases to be free.This follows from a theorem of M. Kahle [21] which states that for p 2 ≥ (3/2 + ǫ) • n −1 • log n the fundamental group π1(XΓ) has property (T) and thus its cohomological dimension is ≥ 2, a.a.s.In the following theorem we describe the range in which the cohomological dimension of π1(XΓ) equals 2.
We wish to mention a recent preprint [3] where random triangular groups are studied; this class of random groups is different from the class of fundamental groups of random clique complexes although these two classes of random groups share several common features.The main result of [3] states that there exists an interval in which the random triangular group is neither free nor possesses the property T. We expect that such intermediate regime exists in the model which we study in this paper.We shall address this issue elsewhere.( Then the fundamental group π1(XΓ) of the clique complex of a random graph Γ ∈ G(n, p) satisfies and in particular π1(XΓ) is torsion free, a.a.s.Moreover, if for some ǫ > 0 one has then gdim(π1(XΓ)) = cd(π1(XΓ)) = 2, (4) a.a.s.
Recall that geometric dimension gdim(G) of a discrete group G is defined as the minimal dimension of an aspherical CW-complex having G as its fundamental group.The cohomological dimension cd(G) is the shortest length of a free resolution of Z viewed as a Z[G]-module.In general cd(G) ≤ gdim(G) and a classical theorem of Eilenberg and Ganea [14] states that cd(G) = gdim(G), except of three lowdimensional cases.At present it is known that the equality cd(G) = gdim(G) holds, except possibly for the case when cd(G) = 2 and gdim(G) = 3.The Eilenberg-Ganea Conjecture states that cd(G) = 2 implies gdim(G) = 2.
The following theorem states that 2-torsion appears in the fundamental group of a random clique complex when we cross the threshold 11/30: where 0 < ǫ < 1/30 is fixed.Then the fundamental group π1(XΓ) has 2-torsion and thus its cohomological dimension and geometric dimension are infinite, a.a.s.
Surprisingly, odd torsion does not appear in fundamental groups of random clique complexes until the triviality threshold p = n −1/3 : Theorem D: [See Theorem 8.1] Let m ≥ 3 be a fixed prime.Assume that where ǫ > 0 is fixed.Then a random graph Γ ∈ G(n, p) with probability tending to 1 has the following property: the fundamental group of any subcomplex Y ⊂ XΓ has no m-torsion.
Surprisingly, we see that for all the assumptions on the probability parameter p considered in Theorems A, B, C, the fundamental groups of random clique complexes have cohomological dimension 1, 2 or ∞, which implies that probabilistically the Eilenberg-Ganea conjecture is satisfied.Note also that in the complementary range, when p = n α with α > −1/3, the clique complex XΓ of a random graph Γ is simply connected a.a.s.(by Theorem 3.4 from [20]) and hence the Eilenberg-Ganea conjecture is also probabilistically satisfied.We may also mention here that any finitely presented group appears as the fundamental group of a clique complex XΓ for a graph Γ ∈ G(n, p) with any n large enough.
Next we state a result in the direction of the Whitehead conjecture.Recall that a connected simplicial complex Y is said to be aspherical if πi(Y ) = 0 for all i ≥ 2; this is equivalent to the requirement that the universal cover of Y is contractible.For 2-dimensional complexes Y the asphericity is equivalent to the vanishing of the second homotopy group π2(Y ) = 0, or equivalently, that any continuous map S 2 → Y is homotopic to a constant map.Random aspherical 2-complexes could be helpful for testing probabilistically the open problems of two-dimensional topology, such as the Whitehead conjecture.This conjecture stated by J.H.C. Whitehead in 1941 claims that a subcomplex of an aspherical 2-complex is also aspherical.Surveys of results related to the Whitehead conjecture can be found in [7], [26].
where ǫ > 0 is fixed.Then, for a random graph Γ ∈ G(n, p), the clique complex XΓ has the following property with probability tending to 1 as n → ∞: any aspherical subcomplex Y ⊂ X (2) Γ satisfies the Whitehead Conjecture, i.e. any subcomplex Y ′ ⊂ Y is also aspherical.
Thus we see that probabilistically, for large finite simplicial complexes the Whitehead conjecture holds for all their aspherical subcomplexes.
We also remark that, as is well known, any finite simplicial complex is homeomorphic to a clique complex XΓ with Γ ∈ G(n, p) for any n large enough; thus every 2-dimensional finite simplicial complex appears up to homeomorphism with positive probability for large n.
Recall that a well known result of Bestvina and Brady [6] states that either the Whitehead conjecture or the Eilenberg-Ganea conjecture must be false.However, Bestvina and Brady consider these two conjectures in a larger class of infinite simplicial complexes and not necessarily finitely presented groups.
We should also point out the limitations of our approach.We have no results in the direction of the Eilenberg-Ganea conjecture near the two critical values of the probability parameter p = n −11/30 and p = n −1/3 .Besides, we do not know the validity of the probabilistic version of the Whitehead conjecture (the analogue of Theorem E) for p ≫ n −1/3 .
A few words about terminology we use in this paper.By a 2-complex we understand a finite simplicial complex of dimension ≤ 2. The i-dimensional simplexes of a 2-complex are called vertices (for i = 0), edges (for i = 1) and faces (for i = 2).
A 2-complex is said to be pure if every vertex and every edge are incident to a face.
The pure part of a 2-complex is the closure of the union of all faces.
The degree of an edge e of X is the number of faces containing e.
The boundary ∂X of a 2-complex X is the union of all edges of degree one.We say that a 2-complex We denote by V (X), E(X), F (X) the sets of vertices, edges and faces of X, correspondingly.We also use the notations We use the notations P 2 for the real projective plane.
The authors thank the referee for making useful critical remarks.

The containment problem
In this section we collect some known results which we shall use in this paper.The only new result here is Theorem 2.8 which describes properties of clean triangulations of surfaces.
Let S be a 2-complex.We have already introduced the notations where v(S) and e(S) denote the number of vertices and edges in S.Although the numbers ν(S) and ν(S) depend only on the 1-skeleton of S, it is convenient to think about ν(S) and ν(S) as being associated to the whole 2-complex S due to the following formula Here In the definition of L(S) the sum is over all the edges e of S and deg(e) is the number of faces containing e.Note that L(S) ≤ 0 assuming that S is closed, i.e. if deg(e) ≥ 2 for every edge e of S.
The embeddability of S into XΓ is equivalent to the embeddability of the 1-skeleton of S into Γ.The following result follows from the well known subgraph containment problem in random graph theory, see [19], Theorem 3.4 on page 56.Theorem 2.1.Let S be a fixed finite simplicial complex.Consider the clique complex XΓ associated to a random Erdős -Rényi graph Γ ∈ G(n, p).Then: ) then the probability that S admits a simplicial embedding into XΓ tends to 0 as n → ∞; ) the the probability that S admits a simplicial embedding into XΓ tends to 1 as n → ∞; Definition 2.2.A graph Γ is said to be balanced if for any proper subgraph A graph Γ is said to be strictly balanced if for any proper subgraph Γ ′ ⊂ Γ one has ν(Γ) < ν(Γ ′ ).
Definition 2.4.A simplicial 2-complex S is called clean if it coincides with the 2-skeleton of the clique complex of its 1-skeleton.
In other words, a triangulation is clean if any clique consisting of three vertices spans a 2-simplex.
Example 2.5.Let Kr+1 be the complete graph on r + 1 vertices.It is easy to see that it is strictly ν-balanced and ν(Kr+1 As a corollary of Theorem 2.1 we obtain: Corollary 2.6.If the probability parameter p satisfies where r ≥ 2 is an integer, then the dimension dim XΓ of a random clique complex XΓ equals r, a.a.s. Example 2.7.Consider a triangulated surface S having a vertex x of degree 3. Clearly such triangulation is not clean.Assume that either S is orientable and has genus > 1 or it is non-orientable and has genus > 2. If Γ denotes the 1-skeleton of S then ν(S) = ν(Γ) < 1/3.Removing the vertex x and the three incident to it edges we obtain a graph Γ ′ ⊂ Γ with v(Γ ′ ) = v(Γ)−1 and e(Γ ′ ) = e(Γ)−3.Since ν(Γ) < 1/3 we see that The following Theorem is analogous to Theorem 27 from [10].
Theorem 2.8.Any clean triangulation of a closed connected surface S with χ(S) ≥ 0 is ν-balanced.Moreover, if χ(S) > 0 then any clean triangulation of S is strictly ν-balanced.
Proof.Let Γ be a graph such that S is the clique complex S = XΓ.Let Γ ′ ⊂ Γ be a proper subgraph and let S ′ = X Γ ′ denotes the clique complex of Γ ′ .Without loss of generality we may assume that Γ ′ is connected.Due to formula (8), the inequality ν(S) < ν(S ′ ) would follow from 3χ(S ′ ) + L(S ′ ) ≥ 3χ(S), (9) since L(S) = 0, e(S) > e(S ′ ) and χ(S) ≥ 0. From now on all homology and cohomology group will have coefficient group Z2 which will be omitted from the notation.Besides, we will use the symbol b ′ i (X) to denote dimZ 2 Hi(X), the i-th Betti number with Z2 coefficients.Consider the exact sequence 0 → H2(S) → H2(S, S ′ ) j * → H1(S ′ ) → H1(S) → H1(S, S ′ ) → 0. ( Here we used that H2(S ′ ) = 0 (since S ′ is a proper subcomplex of S) and H2(S) = Z2.By Poincaré duality, the dimension of H2(S, S ′ ) equals dim H 0 (S −S ′ ) = k, the number of path-connected components of the complement S − S ′ , see Proposition 3.46 from [18].Thus, (10) implies the inequality (9) we see that (9) would follows from (11) once we show that L(S ′ ) ≥ 3k.Note that L(S ′ ) = e1(S ′ ) + 2e2(S ′ ) where ei(S ′ ) denotes the number of edges of S ′ which have degree i, where i = 0, 1.If C1, . . ., C k denote the boundary circles of the connected components of S − S ′ then one has since each edge of S ′ having degree one belong to exactly one of the circles Cj and each edge of degree zero belongs to two circles Cj .Clearly, |Cj | ≥ 3 for each Cj and the inequality L(S ′ ) ≥ 3k follows.
For a triangulation S of a compact orientable surface Σg of genus g one has using the formula (8), Similarly, for a triangulation S of a compact non-orientable surface Ng of genus g one has Thus we see that ν(S) < 1/3 if S is orientable and g > 1 or if S is non-orientable and g > 2.
Remark 2.9.It is easy to show that the assumption χ(S) ≥ 0 of Theorem 2.8 is necessary.More specifically, any closed surface with χ(S) < 0 admits a non-ν-balanced clean triangulation.Indeed, let S be a clean triangulation of a surface with χ(S) < 0; then ν(S) < 1/3 (by ( 12) and ( 13)).Let X ⊂ S be the subcomplex obtained from S by removing an edge e ⊂ S and the interiors of two adjacent to e 2-simplexes.Then tends to 1/3 as r → ∞.Thus, by taking r large enough we shall have ν(S ′ ) > ν(X).The obtained triangulation S ′ is clean and unbalanced since X is a subcomplex of S ′ .
Remark 2.10.Theorem 27 from [10] (which is similar to Theorem 2.8) is valid under an additional assumption χ(S) ≥ 0 which is missing in its statement.The assumption χ(S) ≥ 0 is essential since any closed surface with negative Euler characteristic χ(S) < 0 admits a not µ-balanced triangulation.

Threshold for collapsibility to a graph
In this section we prove Theorem A which we restate below: then, with probability tending to 1 as n → ∞, the clique complex XΓ is simplicially collapsible to a graph, a.a.s.In particular the fundamental group π1(XΓ, x0) of a random clique complex XΓ, where Γ ∈ G(n, p), is free, for any choice of the base point x0 ∈ XΓ.Moreover, under the above assumptions each connected component of the 2-skeleton X (2) Γ is homotopy equivalent to a wedge of circles and 2-spheres, a.a.s.
The proof of Theorem 3.1 uses a deterministic combinatorial assertion described below as Theorem 3.2.In its statement we use the notation where S is a simplicial 2-complex and v(S) and e(S) denote the number of its vertices and edges.We will also use the invariant where S runs over all subcomplexes of X.
We shall denote by S1 the tetrahedron (the 2-complex homeomorphic to the sphere S 2 and having 4 vertices, 6 edges and 4 faces) and by S2 the triangulation of S 2 having 5 vertices, 9 edges and 6 faces.Clearly, ν(S1) = 2/3 > 1/2 and ν(S2) = 5/9 > 1/2.The complexes S1 and S2 play a special role in our study: Theorem 3.2 below implies that any closed 2-complex X satisfying ν(X) > 1/2 contains either S1 or S2 as a simplicial subcomplex.Theorem 3.2.There exists an infinite set L of isomorphism types of finite simplicial 2-complexes satisfying the following properties: (1) for any S ∈ L one has ν(S) ≤ 1/2; (2) the set L has at most exponential size in the following sense: for an integer E let LE denote the set {S ∈ L; e(S) ≤ E}.Then for some positive constants A and B one has where A and B are independent of E; (3) any closed pure 2-complex X contains a simplicial subcomplex isomorphic to some S ∈ L ∪ {S1, S2}.
Property (3) is the main universal feature of the set L.
Proof of Theorem 3.2.We start with a few remarks: For a triangulated 2-disc X having v vertices such that among them there are vi internal vertices, one has Thus one has ν(X) = 1/2 for vi = 3 and ν(X) > 1/2 only for vi = 0, 1, 2. Formula (17) follows from the relations 3f = 2e − v ∂ and v − e + f = 1 where v ∂ = v − vi is the number of vertices on the boundary.The operation of adding an external edge to a simplicial complex X gives a simplicial complex X ′ = X ∪ E where E is a arc (i.e. a space homeomorphic to [0, 1]) and If X is obtained from a triangulated disc with vi internal vertices by adding c external edges, then and therefore we see that ν(X) ≤ 1/2 if and only if vi + c ≥ 3.
We denote by L the set of isomorphism types of finite simplicial 2-complexes S having the following properties: the pure part S0 of S admits a surjective simplicial map f : S ′ → S0 where: (a) S ′ is a triangulated disc with one internal vertex; (b) f is bijective on the set of faces; (c) the image of any edge of S ′ is an edge of S0; (e) the complex S is obtained from its pure part S0 by adding at most 2 external edges; (f) and finally we require that Typical examples of complexes from L are given below.
Next we show that the set L satisfies property (2) of Theorem 3.2.According to W. Brown [8], the number of isomorphism types of triangulations of the disc S ′ with v vertices having one internal vertex is less than or equal to 2v here we use formula (4.7) from [8] with v = m+4 and n = 1.This implies that the number of isomorphism types of triangulations of the disc S ′ with at most v vertices and one internal vertex is less than or equal to We want to estimate above the number of elements S ∈ L satisfying e(S) ≤ E. For S ∈ L with e(S) ≤ E, let f : S ′ → S0 be a surjective simplicial map as in the definition of L.Here S0 is the pure part of S and S is obtained from S0 by adding c = 0, 1, 2 edges.Then using (19), we find The complex S0 is obtained from S ′ by identifying at most 2 pairs of vertices or by identifying a triple of vertices; the identification of vertices determines the identification of edges.As we noted above, there are 19 types of quotients.Hence we obtain (assuming that E ≥ 6) In the above inequality the first factor (E/2 + 2) 4 accounts for the ways of doing identifications of vertices and the second factor (E/2+2) 4 accounts for the ways to add 2 additional edges.Since (E/2+2) 4 ≤ 4 E/2+2 we see that This proves that the set L has property (2) of Theorem 3.  Note that a triangulated disc with k internal points may be obtained as a quotient of a triangulated disc with k − 1 internal points by identifying two vertices and two adjacent edges on the boundary.This fact is illustrated by Figure 2.
Example 3: Consider a simplicial surjective map f : X ′ → X where X ′ is a triangulated disc and f is bijective on faces and every edge of X ′ is mapped to an edge of X and such that v(X ′ ) − v(X) = 1 and e(X ′ ) − e(X) = 1, i.e. exactly two vertices and two (adjacent) edges are identified.If X ′ has i internal vertices then we call such an X a scroll with i internal points.A scroll with two internal points is an element of L. In particular, every triangulated disc with 3 internal points belongs to L.
Example 4: A scroll with one internal point and with one external edge added is an element of L.
Example 5: As above, consider a simplicial surjective map f : X ′ → X where X ′ is a triangulated disc and f is bijective on faces and every edge of X ′ is mapped to an edge of X. Assume that exactly two pairs of vertices and two pairs of adjacent edges are identified, i.e. v(X ′ )−v(X) = 2 and e(X ′ )−e(X) = 2.If X ′ has one internal vertex then X ∈ L. We shall call such an X disc with one internal point and with two scrolls.Now we show that the set L has property (3) of Theorem 3.2.We shall assume the negation of property (3) and arrive to a contradiction.Hence, below we assume that there exists a closed pure 2-complex X which contains no subcomplexes isomorphic to any S ∈ L ∪ {S1, S2}.
Consider a vertex v ∈ X and let LkX (v) be the link of v in X; it is a graph having no univalent vertices (since X is closed) and hence each connected component of LkX(v) contains a simple cycle C ⊂ LkX (v).The cone D = vC ⊂ X with base C and apex v is a disc with one internal point.
There may exist at most one external edge, i.e. an edge e ⊂ X such that e ⊂ D and ∂e ⊂ D (since otherwise the union of D and of two such edges would be isomorphic to an element of L, see Example 1).Consider a vertex w ∈ C = ∂D which is not incident to an external edge (such point exists since C has at least 3 vertices).Let w ′ , w ′′ ∈ C be the two neighbours of w along C. The link LkX (w) of the vertex w in X is a graph without univalent vertices and the link α = LkD(w) is an arc connecting the points w ′ and w ′′ .It is obvious that the arc α is contained in a subgraph Γ ⊂ LkX (w) which is homeomorphic either to the circle or to one of the two graphs shown in Figure 4 (the graph Γ can be obtained by extending α in LkX (w) until the extension "hits itself").Thus the complex X contains the cone wΓ over Γ with apex w.The intersection wΓ ∩ D clearly contains wα, the cone over the arc α.Any vertex u of (wΓ ∩ D) − wα corresponds to an edge e ⊂ X such that ∂e = {w, u} and e ⊂ D. By construction, we know that there are no such external edges.Thus, we see that the set of vertices of wΓ ∩ D coincides with the set of vertices of wα.
In the case when Γ is homeomorphic to one of the graphs shown in Figure 4 the union wΓ ∪ D ⊂ X is a disc with one internal point and with two scrolls which is impossible due to Example 5. Thus the only remaining possibility is that Γ is a simple circle.The union wΓ ∪ D can be the tetrahedron S1 (iff Γ − Int(α) is a single edge contained in ∂D); otherwise the union wΓ ∪ D is a disc.The first possibility contradicts our assumptions (we know that X does not contain S1 as a subcomplex), therefore the union D1 = wΓ ∪ D ⊂ X is a disc with two internal points v, w.
Next we repeat the above arguments applied to D1 ⊂ X instead of D ⊂ X.Consider a point w1 ∈ ∂D1 and its two neighbours w ′ 1 , w ′′ 1 ∈ C1 = ∂D1.The link LkX (w1) is a graph without univalent vertices and α1 = LkD(w1) is an arc connecting the points w ′ 1 , w ′′ 1 .The arc α1 is contained in a subgraph Γ ⊂ LkX (w1) which is homeomorphic either to the circle or to one of the graphs shown in Figure 4.The set of vertices of Γ contained in D1 coincides with the set of vertices of α1 (since otherwise X would contain a disc with two internal points and with one external edge which contradicts Example 2).In the case when Γ is homeomorphic to one of the graphs of Figure 4 the union w1Γ ∪ D1 ⊂ X contains a scroll with two internal points which is impossible because of Example 3. If Γ is a simple circle then the union w1Γ ∪ D1 is either S2 (iff Γ − Int(α1) is a single edge contained in ∂D1), or the union D2 = w1Γ ∪ D1 is a disc with 3 internal points v, w, w1.Both these possibilities contradict our assumptions concerning X.This completes the proof.
Proof of Theorem 3.1.Consider a random graph Γ ∈ G(n, p) and its clique complex XΓ.Clearly, XΓ is connected if and ony if Γ is connected.Since p ≪ n −1/2 we know that dim XΓ ≤ 3 a.a.s.; see Corollary 2.6.The 3-simplexes of XΓ are in one-to-one correspondence with the embedding of the complete graph K4 into Γ.Let us show that each 3-simplex of XΓ has at least three free faces.Indeed, assume that there is a 3-simplex in XΓ with less than three free faces.Then the complex S formed as the union S = S1 ∪ S2 ∪ S3 of three tetrahedra S1, S2, S3, where the intersections S1 ∩ S2 and S1 ∩ S3 are 2-simplexes and S2 ∩ S3 is an edge, would be embeddable into XΓ; however this is impossible due to Theorem 2.1 since ν(S) = 6/12 = 1/2 and p ≪ n −1/2 .Choosing a free face in each 3-simplex and performing collapse XΓ ց X ′ Γ we obtain a 2-complex X ′ Γ .Clearly, X ′ Γ does not contain S1 and S2 as subcomplexes.Next we perform a sequence of simplicial collapses where on each step we collapse all free faces of 2-simplexes.After finitely many such collapses we obtain a complex X ∞ Γ which is either (a) a graph, or (b) a closed 2-dimensional simplicial complex.We know that X ∞ Γ contains neither S1 nor S2 as a subcomplex, and besides, π1(XΓ, x0) = π1(X ∞ Γ , x0) for any base point x0.
We claim that option (b) happens with probability tending to zero as n → ∞; in other words, X ∞ Γ is a graph, a.a.s.Indeed, by Theorem 3.2 if X ∞ Γ is not a graph then it admits a simplicial embedding S → X ∞ Γ of some S ∈ L. However for a fixed S ∈ L one has and therefore (using Theorem 3.2) the probability that X ∞ Γ is not a graph is less than or equal to as n → ∞ since we assume that pn 1/2 → 0.

Uniform hyperbolicity
Let X be a finite simplicial complex.For a simplicial loop γ : S 1 → X (1) ⊂ X we denote by |γ| the length of γ.If γ is null-homotopic, γ ∼ 1, we denote by AX(γ) the area of γ, i.e. the minimal number of triangles 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 and the mapping cylinder of i is a disc with boundary S 1 × 0, see [4].
Clearly I(X) = I(X (2) ) i.e. the isoperimetric constant I(X) depends only on the 2-skeleton X (2) .Define the following invariant of X The inequality I(X) ≥ a means that for any null-homotopic loop γ in X one has the isoperimetric inequality AX(γ) ≤ a −1 • |γ|.The inequality I(X) < a means that there exists a null-homotopic loop γ in X with AX (γ) > a −1 • |γ|, i.e. γ is null-homotopic but does not bound a disk of area less than a −1 • |γ|.
It is well known that I(X) > 0 if and only if π1(X) is hyperbolic in the sense of M. Gromov [16].
It is known that the number I(X) coincides with the infimum of the ratios |γ| • AX(γ) −1 where γ runs over all null-homotopic simplicial prime loops in X, i.e. such that their lifts to the universal cover X of X are simple.Note that any simplicial filling S 1 i → V b → X for a prime loop γ : S 1 → X has the property that V is a simplicial disc and i is a homeomorphism i : S 1 → ∂V .Hence for prime loops γ the area AX(γ) coincides with the minimal number of 2-simplexes in any simplicial spanning disc for γ.
The following Theorem 4.2 gives a uniform isoperimetric constant for random complexes XΓ where Γ ∈ G(n, p).It is a slightly stronger statement than simply hyperbolicity of the fundamental group of Y .Theorem 4.2.Suppose that for some ǫ > 0 the probability parameter p satisfies Then there exists a constant cǫ > 0 depending only on ǫ such that the clique complex XΓ of a random graph Γ ∈ G(n, p), with probability tending to 1 as n → ∞, has the following property: any subcomplex Y ⊂ XΓ satisfies I(Y ) ≥ cǫ; in particular, for any subcomplex Y ⊂ XΓ the fundamental group π1(Y ) is hyperbolic, a.a.s.
The proof of Theorem 4.2 is given in the Appendix at the end of the paper.
If L(S) = 0 then each edge has degree 2 and S is a pseudo-surface.Using Corollary 2.1 from [12] we obtain that S is a genuine triangulated surface without singularities and the only surface satisfying b1(S) = b2(S) = 0 is the projective plane.
The case L(S) = −1 is impossible.Indeed, if L(S) = −1 then there is a single edge of degree 3 and all other edges have degree 2. The link of a vertex incident to the edge of degree 3 will be a graph with all vertices of degree 2 and one vertex of degree 3 which is impossible.
Assume now that L(S) = −2.There are two possibilities: (a) either there are two edges of degree 3 and all other edges have degree 2, or (b) there is a single edge of degree 4 and all other edges have degree 2.
The possibility (a) cannot happen.Indeed, if e, e ′ are two edges of degree 3 and v is a vertex incident to e but not to e ′ then the link of v is a graph with all vertices of degree 2 and one vertex of degree 3 which is impossible.
Consider now the case (b).Let e be the edge of degree 4 and let v, w be the endpoints of e. Repeating the arguments of the proof of Theorem 2.4 from [12] (see Case C in [12]) we see that S is obtained from a pseudo-surface S ′ by identifying two adjacent edges.Since S ′ and S are homotopy equivalent, we obtain that b1(S ′ ) = b2(S ′ ) = 0. Using Corollary 2.1 from [12] and classification of surfaces we see that S ′ is homeomorphic to the projective plane; therefore S is isomorphic to a simplicial complex of type P ′ as explained above.
Corollary 5.2.Let S be a connected 2-complex with b2(S) = 0.If ν(S) > 1/3 then S is homotopy equivalent to a wedge of circles and projective planes.Proof.Let σ ⊂ Z be an arbitrary face.Starting with the complex Z − Int(σ) and collapsing subsequently faces across the free edges we shall arrive to a connected graph Γ (due to our assumption about the absence of closed subcomplexes).Let us show that b1(Γ) ≤ 1.The inequality ν(Z) > 1/3 is equivalent to 3χ(Z) + L(Z) > 0 (see formula (8)) where L(Z) ≤ 0 (since Z is closed) and hence χ(Z) ≥ 1. Therefore χ(Γ) = χ(Z)−1 ≥ 0 which implies b1(Γ) ≤ 1.Hence, Γ is either contractible or it is homotopy equivalent to the circle.In the first case, Z is homotopy equivalent to S 2 .In the second case, Z is homotopy equivalent to the result of attaching a 2-cell to the circle, S 1 ∪ f e 2 .Since b2(Z) = 1 we obtain that deg(f ) = 0, and hence Z is homotopy equivalent to S 1 ∨ S 2 .We see that the inclusion ∂σ → Z − Int(σ) ≃ Γ is homotopically trivial in both cases.
Lemma 5.7.Let Z be a minimal cycle of type B such that ν(Z) > 1/3.Suppose that any edge e of Z has degree ≤ 3. Then Z is isomorphic (as a simplicial complex) to the union P 2 ∪ D 2 , where P 2 and D 2 are triangulated projective plane and the disc, P 2 ∩ D 2 = ∂D 2 = P 1 ⊂ P 2 , and the loop ∂D 2 has either 3, 4 or 5 edges.Here P 1 ⊂ P 2 denotes a simple homotopically nontrivial simplicial loop on the projective plane.In particular, Z is homotopy equivalent to S 2 and for any face σ ⊂ P 2 ⊂ Z the boundary ∂σ is null-homotopic in Z − Int(σ).
Proof.Let Z ′ be a strongly connected proper closed 2-dimensional subcomplex of Z. Since any edge of Z ′ has degree ≤ 3 in Z ′ , it follows from Lemma 5.1 that Z ′ is homeomorphic to P 2 .
Denote Z ′′ = (Z − Z ′ ) and let G be the graph G = Z ′ ∩ Z ′′ .Let Γ be the subgraph of the 1-skeleton Z (1) of Z formed by the edges of degree 3 in Z. Clearly G ⊂ Γ.By definition of Γ and the assumptions of the Lemma, any edge of Γ has degree 3 in Z and every edge of Z which is not in Γ must have degree 2 in Z.In particular one has that L(Z) = −e(Γ).
The graph G (and therefore Γ) must contain a cycle, since otherwise Z is homotopy equivalent to Z ′ ∨ Z ′′ and thus b2(Z ′′ ) = 1, contradicting the minimality of Z.In particular e(G) ≥ 3.Moreover, Γ has at most 5 edges since ν(Z) > 1/3 implies L(Z) ≥ −5 (using formula (8) and L(Z) = −e(Γ) ≤ −e(G)).Hence Γ either contains exactly one cycle (of length 3, 4 or 5) or Γ is a square with one diagonal.Let us show that the latter case is impossible.Indeed, suppose that Γ is a square with one diagonal.Let v0 be one of the vertices of degree 3 in Γ.Then v0 is incident to exactly three odd degree edges in Z (corresponding to the three neighbours v1, v2, v3 of v0 in Γ).In particular the link LkZ (v0) would have an odd number of odd degree vertices which is impossible.We conclude that b1(Γ) = b1(G) = 1.
We now show that Γ is a cycle and therefore G = Γ.Suppose that Γ contains an edge e with a free vertex v. Then the link LkZ (v) is a graph with exactly one vertex of degree 3 and all other vertices of degree 2. This contradicts the fact that every graph has an even number of odd degree vertices.
We have shown that G = Γ is a cycle of length 3, 4 or 5 and that all edges of G have degree 3 in Z and all edges of Z which are not in G have degree 2. Recall that Z = Z ′ ∪G Z ′′ where Z ′ is a triangulated projective plane.Since for any edge e ∈ G, one has deg Z ′′ (e) = degZ(e) − deg Z ′ (e) = 1 it follows that Z ′′ is a pseudo-surface with boundary.Moreover, since χ(G) = 0 and χ(Z ′ ) = 1 we obtain which would contradict the minimality of Z. Hence we see that Z is homotopy equivalent to S 2 and any 2-simplex σ ⊂ Z ′ has the required property.Lemma 5.8.Let Z be a minimal cycle of type B such that ν(Z) > 1/3 and such that an edge e of Z has degree ≥ 4. Then Z is isomorphic (as a simplicial complex) to the quotient q : Ẑ = P 2 ∪ D 2 → Z of a minimal cycle Z of type B with ν( Ẑ) > 1/3 and such that all edges of Z have degree ≤ 3 (as described in the previous Lemma); the map q identifies two vertices and two adjacent edges.In particular, Z is homotopy equivalent to S 2 and for any face σ ⊂ q(P 2 ) ⊂ Z the boundary ∂σ is contractible in Z − Int(σ).
Proof.Let Γ be the subgraph of the 1-skeleton of Z which is the union of the edges of degree ≥ 3.As in the proof of the previous lemma, the inequality ν(Z) > 1/3 implies L(Z) ≥ −5 and using our assumption that at least one edge of Γ has degree ≥ 4 we obtain −5 ≤ L(Z) ≤ −e(Γ) − 1, i.e.Γ has at most 4 edges.On the other hand e(Γ) ≥ 3 since Γ must contain a cycle as follows from the argument used in the proof of Lemma 5.7.Thus we have consider the cases e(Γ) equals 3 or 4.
Define Γ odd to be the subgraph of Γ formed by the edges of odd degree in Z.The graph Γ odd is non-empty; indeed, since e(Γ) ≥ 3 and every edge of Γ with even degree must have degree ≥ 4, the assumption Γ odd = ∅ would imply L(Z) ≤ −2e(Γ) ≤ −6 contradicting L(Z) ≥ −5.Furthermore the graph Γ odd may not have a free vertex.If Γ odd contained an edge e with a free vertex v then the link LkZ(v) would be graph with exactly one vertex of odd degree contradicting the fact that every graph has an even number of odd degree vertices.We obtain in particular that e(Γ odd ) ≥ 3 and b1(Γ odd ) ≥ 1.
We can now describe the graph Γ.
If e(Γ) = 3, then all edges of Γ must have odd degree in Z, i.e.Γ = Γ odd .Furthermore, since L(Z) ≥ −5 and Z has at least one edge of degree > 3, it follows that Γ is a cycle formed by two edges of degree 3 and one edge of degree 5.In particular, L(Z) = −5.Denote the edge of degree 5 by e.Let v be a vertex of e. Then the link LkZ(v) is a graph with exactly two vertices of odd degree.One of these vertices has degree 3 in the link LkZ(v) and the other vertex has degree 5.The link Lk(v) is connected since otherwise we would have b1(Z) ≥ 1 (by Corollary 2.1 from [12]) implying χ(Z) ≤ 1 and L(Z) ≥ −2, a contradiction.Hence, the link Lk(v) is a connected graph with one vertex of degree 3, one vertex of degree 5 and all other vertices of degree 2. There are two possibilities for Lk(v) which are shown in Figure 6.A neighbourhood of the point v is the cone v • Lk(v) over the link Lk(v).We may represent Lk(v) as the union A ∪ B where A is a circle and the intersection A ∩ B is one point, the vertex of degree 5. We may cut Z from the vertex v and along the edge e introducing instead of v two new vertices v1 and v2 end two edges (of degree 3 and 2) instead of e. Formally we replace the cone v • Lk(v) by the union of two cones (v1 • A) ∪ (v2 • B) as shown in Figure 7.The obtained 2-complex Ẑ is a minimal cycle χ( Ẑ) = χ(Z) = 2 and L( Ẑ) = −3.To apply Lemma 5.7 we want to show that ν( Ẑ) > 1/3.The negation ν( Ẑ) ≤ 1/3 means that there exists a subgraph H ⊂ Ẑ(1) with ν(H) ≤ 1/3.Identifying two adjacent edges of H we obtain a subgraph H ′ of the 1-skeleton Z (1) with v(H ′ ) = v(H) − 1, e(H ′ ) = e(H) − 1 and now the inequality e(H) ≥ 3v(H) implies e(H ′ ) ≥ 3v(H ′ ), and therefore ν(Z) ≤ 1/3 which contradicts our assumption ν(Z) > 1/3.
From Lemma 5.7 we know that Ẑ is isomorphic to P 2 ∪ D 2 where the intersection P 2 ∩ D 2 = P 1 ⊂ P 2 has length 3 (since L( Ẑ) = −3).Therefore, we obtain that Z can be obtained from P 2 ∪ D 2 by identifying two adjacent edges.
Consider now the remaining case e(Γ) = 4. Then Γ has three edges of degree 3 and one edge of degree 4. Besides, the edges of degree 3 form a cycle since b1(Γ odd ) ≥ 1. Suppose e(Γ) = 4, i.e.Γ = Γ odd ∪ e where Γ odd is a cycle of length 3 with all edges of degree 3, and where e is the edge of degree 4. Then e contains a free vertex v in Γ.Since deg Z (e) = 4, we see that the link LkZ(v) is topologically a wedge of two circles.A neighbourhood of v in Z is a cone v • Lk(v) and representing the link Lk(v) and a union of two circles A ∪ B intersecting at the vertex of degree 4, we may replace the cone v • Lk(v) by the union of two cones (v1 • A) ∪ (v2 • B) where v1 and v2 are two new vertices.We obtain a simplicial complex Ẑ such that Z is obtained from Ẑ by identifying two adjacent edges.Clearly Ẑ is a minimal cycle of type B with all edges of degree ≤ 3.As in the case e(Γ) = 3 considered above one shows that ν( Ẑ) > 1/3.Thus, we see that Ẑ is a minimal cycle satisfying conditions of Lemma 5.7 and Z is obtained from Ẑ by identifying two adjacent edges.
Corollary 5.9.Let X be a connected 2-complex satisfying ν(X) > 1/3.Then X is homotopy equivalent to a wedge of circles, 2-spheres and real projective planes.Besides, there exists a subcomplex X ′ ⊂ X containing the 1-skeleton of X and having the homotopy type of a wedge of circles and real projective planes and such that π1(X ′ ) → π1(X) is an isomorphism.In particular, the fundamental group of X is a free product of several copies of Z and Z2 and hence it is hyperbolic.This Corollary is equivalent to Theorem 1.2 from [5].The proof given below is independent of the arguments of [5].Our proof is based on the classification of minimal cycles described above in the this section.This classification of minimal cycles is not only useful for the proof of Corollary 5.9 but it is also plays an important role in the proofs of many results presented in this paper.
Proof.We will act by induction on b2(X).
If b2(X) = 0 and ν(X) > 1/3 then using Corollary 5.2 we see that the complex X is homotopy equivalent to a wedge of circles and projective planes.In this case one sets X ′ = X and the result follows.
by the induction hypothesis there exists a subcomplex Y ′ ⊂ Y such that π1(Y ′ ) → π1(Y ) is an isomorphism and Y ′ is homotopy equivalent to a wedge of circles and projective planes.However X is homotopy equivalent to Y ∨ S 2 and the result follows (with X ′ = Y ′ ).

The Whitehead Conjecture
If p ≪ n −1/3 then dim XΓ ≤ 5 a.a.s.(see Corollary 2.6).We consider below the 2-dimensional skeleton X (2) Γ which can be viewed as a random 2-complex.In this section we shall examine the validity of the Whitehead Conjecture for aspherical subcomplexes of X Γ .Recall that for any simplicial complex K, its first barycentric subdivision K ′ is a clique complex.Thus, if there exists a counterexample to the Whitehead Conjecture then there exists a counterexample of the form X (2) Γ for certain graph Γ. Theorem 6.1.Assume that p ≪ n −1/3−ǫ , where ǫ > 0 is fixed.Then, for a random graph Γ ∈ G(n, p) the 2-skeleton X (2) Γ of the clique complex XΓ has the following property with probability tending to 1 as is aspherical if and only if every subcomplex S ⊂ Y having at most 2ǫ −1 edges is aspherical.
Intuitively, this statement asserts that a subcomplex Y ⊂ X (2) Γ is aspherical iff it has no "small bubbles" where by a "bubble" we understand a subcomplex S ⊂ Y with π2(S) = 0 and "a bubble is small" if it satisfies the condition e(S) ≤ 2ǫ −1 .Corollary 6.2.Assume that p ≪ n −1/3−ǫ , where ǫ > 0 is fixed.Then, for a random graph Γ ∈ G(n, p), the clique complex XΓ has the following property with probability tending to 1 as n → ∞: any aspherical subcomplex Y ⊂ X (2) Γ satisfies the Whitehead Conjecture, i.e. any subcomplex Y ′ ⊂ Y is also aspherical.
Here is another interesting statement about the local structure of aspherical subcomplexes of X (2) Γ .Corollary 6.3.Assume that p ≪ n −1/3−ǫ , where ǫ > 0 is fixed.Then, for a random graph Γ ∈ G(n, p) the clique complex XΓ has the following property with probability tending to 1 as n → ∞: for any aspherical subcomplex Y ⊂ X We now start preparations for the proofs of theorems 6.1 and 6.3 which appear below in this section.Corollary 6.2 obviously follows from Theorem 6.1.
Let Y be a simplicial complex with π2(Y ) = 0.As in [12], 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 .
We define M (Y ) = 0, if π2(Y ) = 0. Lemma 6.4 (See Corollary 5.3 in [12]).Let Y be a 2-complex with Combining this with Theorem 4.2 we obtain: Lemma 6.5.Assume that the probability parameter satisfies p ≪ n −1/3−ǫ where ǫ > 0 is fixed.Then there exists a constant Cǫ > 0 such that for a random graph Γ ∈ G(n, p) the clique complex XΓ has the following property with probability tending to one: for any subcomplex Y ⊂ X (2) Γ one has M (Y ) ≤ Cǫ.
Clearly, Lemma 6.5 follows from Theorem 4.2 and from Lemma 6.4.
Proof of Theorem 6.1.Let Γ be a random graph, Γ ∈ G(n, p), p ≪ n −1/3−ǫ , and let Y ⊂ X (2) Γ be a 2-dimensional subcomplex.Suppose that π2(Y ) = 0, i.e.Y is not aspherical.Using Lemma 6.5 we have M (Y ) ≤ Cǫ a.a.s.where Cǫ > 0 is a constant depending on ǫ.There is a homotopically nontrivial simplicial map φ : S → Y where S is a triangulation of the sphere S 2 having at most Cǫ faces.Hence, Y must contain as a subcomplex a simplicial quotient S ′ = φ(S) of a triangulation S of the sphere S 2 having at most Cǫ faces and such that φ * : π2(S) → π2(S ′ ) is nonzero.
If b2(S) = 0 then by Lemma 5.1 there is a subcomplex K ⊂ S which is homotopy equivalent to the real projective plane P 2 .By a Theorem of Crockfort [9], see also [1], an aspherical complex cannot contain such K as a subcomplex; hence π2(Y ) = 0, i.e.Y is not aspherical.
We want to show that each S ⊂ Y ⊂ X Γ , e(S) ≤ 2ǫ −1 is collapsible to a graph.Indeed, performing all possible simplicial collapses on S we either obtain a graph or a closed 2-dimensional complex S ′ with e(S ′ ) ≤ 2ǫ −1 and ν(S ′ ) > 1/3.If b2(S ′ ) > 0 then S ′ contains a minimal cycle Z ⊂ S ′ , ν(Z) > 1/3 and using Lemma 5.6 we see that S ′ is not aspherical -a contradiction.If b2(S ′ ) = 0 then by Lemma 5.1 we see that S ′ contains a subcomplex X ⊂ S ′ homotopy equivalent to P 2 and S ′ is not aspherical by a theorem of Cockcroft [9].Hence the only possibility is that S is collapsible to a graph.
Then the fundamental group π1(XΓ) of the clique complex of a random graph Γ ∈ G(n, p) has geometric dimension and cohomological dimension at most 2, and in particular π1(XΓ) is torsion free, a.a.s.Moreover, if then the geometric dimension and the cohomological dimension of π1(XΓ) equal two.
Proof of Theorem 7.1.Consider the set C60 of isomorphism of simplicial complexes having at most edges, where cǫ > 0 is the constant given by Theorem 4.2 for ǫ = 1/30.This set is clearly finite.For any n, consider the set Xn of graphs Γ ∈ G(n, p) such that the corresponding clique complex XΓ does not contain as a subcomplex complexes S ∈ C60 satisfying ν(S) ≤ 11/30 and such that for any subcomplex Y ⊂ XΓ one has I(Y ) ≥ cǫ.From Theorem 2.1 and Theorem 4.2 we know that, under our assumption p ≪ n −11/30 , the probability of this set Xn of graphs tends to one as n → ∞.
Here we use the results of Eilenberg and Ganea [14] in conjunction with the theorem of Swan [27] stating that a group of cohomological dimension one is a free group.The equality cd(π1(XΓ)) = 2 under the assumptions n −1/2 ≪ p ≪ n −11/30 follows from the result of [21], Theorem 1.2 which states that for the fundamental group π1(XΓ) has property T (a.a.s.) implying cd(π1(XΓ)) > 1.
Consider the minimal cycles Z ∈ C60 and their all possible embeddings Z ⊂ XΓ where Γ ∈ Xn.By Lemma 5.6 each such Z contains a 2-simplex σ such that ∂σ is null-homotopic in Z − Int(σ).We remove subsequently one such 2-simplex from each of the minimal cycles Z ⊂ XΓ.The union of the 1-skeleton of XΓ and the remaining 2-simplexes is a 2-complex which we denote by YΓ.Clearly π1(YΓ) = π1(XΓ).To show that YΓ is aspherical we shall apply Theorem 6.1.We need to show that any subcomplex S ⊂ YΓ, where S ∈ C60, is aspherical.By the above construction we know that S ⊂ YΓ cannot contain minimal cycles, and therefore b2(S) = 0. Without loss of generality we may assume that S is closed, pure and strongly connected; then Lemma 5.1 implies that S must contain a triangulation of the projective plane or its quotient with two adjacent edges are identified.
We obtain that only triangulations S of P 2 having less than 30 edges, e(S) < 30, are embeddable into XΓ where Γ ∈ Xn.
Recall that a triangulation of a 2-complex is called clean if for any clique of three vertices {v0, v1, v2} the complex contains also the simplex (v0v1v2).We shall use the following fact: any clean triangulation of the projective plane P 2 contains at least 11 vertices and 30 edges, see [17].The minimal clean triangulation is shown in Figure 8; the antipodal points of the circle must be identified.Any triangulation S of P 2 containing less than 30 edges is not clean, i.e. it contains a cycle of length 3 which is not filled by a triangle.If this cycle is null-homologous than we may split S into two smaller surfaces one of which is a disk and another is a projective plane with smaller number of edges.Continuing by induction, we obtain that for any triangulation S of P 2 containing less than 30 edges there is a cycle of length 3 representing a non-contractible loop in S.
We claim that YΓ contains no subcomplexes S with e(S) ≤ 60 which are triangulations of P 2 .Indeed, if S is embedded into YΓ, where Γ ∈ Xn, then a nontrivial cycle of S bounds a triangle in XΓ.In particular, the inclusion S → XΓ induces a trivial homomorphism of the fundamental groups π1(S) → π1(XΓ).Since the inclusion induces an isomorphism π1(YΓ) → π1(XΓ) we obtain that the inclusion S ⊂ YΓ also induces a trivial homomorphism π1(S) → π1(YΓ) however now the length 3 cycle of S may bound a larger disc and not a simple 2-simplex.We may apply Theorem 4.2 about uniform hyperbolicity to estimate the size of the minimal bounding disc for this cycle.Since I(YΓ) ≥ cǫ where ǫ = 1/30, we see that the area of the bounding disc is ≤ 3c −1 ǫ .We obtain that there exists a subcomplex S ⊂ L ⊂ YΓ such that π1(S) → π1(L) is trivial and Since Γ ∈ Xn we see that ν(L) > 1/3.By construction, L (as well as YΓ) may not contain minimal cycles since any minimal cycle Z satisfying ν(Z) > 11/30 must have at most 60 edges; therefore b2(L) = 0. We may assume that L is strongly connected and pure.Then by Lemma 5.1 we see that each strongly connected pure component of L must be isomorphic either to the projective plane or to its quotient, and in both cases we obtain a contradiction to the homomorphism π1(S) → π1(L) being trivial.
Similarly, one shows that YΓ contains no subcomplexes S ′ isomorphic to the quotients of a triangulation of P 2 with two adjacent edges identified and with e(S ′ ) ≤ 60.One has and for Γ ∈ Xn we shall find subcomplexes S ′ ⊂ XΓ only if e(S ′ ) < 10.Thus, using the result of [17], we obtain that that if S ′ is embedded into XΓ, where Γ ∈ Xn, then there is a cycle of length 3 in S ′ which is not null-homotopic in S ′ ; this cycle bounds a triangle in XΓ and as a result the inclusion S ′ → XΓ induces a trivial homomorphism of the fundamental groups π1(S ′ ) → π1(XΓ).Repeating the arguments of the preceding paragraph we find a subcomplex S ′ ⊂ L ⊂ YΓ such that π1(S ′ ) → π1(L) is trivial and L satisfies (25).As above we find that ν(L) > 1/3, b2(L) = 0 and therefore L is an iterated wedge of projective planes or projective planes with two adjacent edges identified; this contradicts the fact that π1(S ′ ) → π1(L) is trivial.
7.1 Proof of Theorem 7.2

The number of combinatorial embeddings
Consider two 2-complexes S1 ⊃ S2.Denote by vi and ei the numbers of vertices and faces of Si.We have v1 ≥ v2 and e1 ≥ e2.We will assume that e1 > e2.
Let ν(S1, S2) denote the ratio From Corollary 5.9 we know that the fundamental group of any 2-complex ν(X) > 1/3 is a free product of several copies of Z and Z2 and has no m-torsion, as we assume that m ≥ 3. Since the fundamental group of any X ∈ Xm has m-torsion, where m ≥ 3, one has ν(X) ≤ 1/3 for any X ∈ Xm.Hence, using the finiteness of Xm and the results on the containment problem (Theorem 2.1) we see that for p ≪ n −1/3−ǫ the probability that a random complex XΓ, where Γ ∈ G(n, p), contains a subcomplex isomorphic to one of the complexes X ∈ Xm tends to 0 as n → ∞.Hence, we obtain that (a.a.s.) any subcomplex Y ⊂ XΓ does not contain X ∈ Xm as a subcomplex and therefore the fundamental group of Y has no m-torsion.
A Appendix: Proof of Theorem 4.2 In this Appendix we give a complete and self-contained proof of Theorem 4.2 which plays a key role in this paper.As we mentioned above, this statement is closely related to Theorem 1.1 from [5].The proof of Theorem 4.2 given below is similar to the arguments of [4], [5] and [12] and is based on two auxiliary results: (1) the local-to-global principle of Gromov [16] and on (2) Theorem A.2 giving uniform isoperimetric constants for complexes satisfying ν(X) ≥ 1/3 + ǫ.
The local-to-global principle of Gromov can be stated as follows: Theorem A.1.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 Let X be a 2-complex satisfying ν(X) > 1/3.Then by Corollary 5.9 the fundamental group of X is hyperbolic as it is a free product of several copies of cyclic groups Z and Z2.Hence, I(X) > 0. The following theorem gives a uniform lower bound for the numbers I(X).
Theorem A.2.Given ǫ > 0 there exists a constant Cǫ > 0 such that for any finite pure 2-complex X with ν(X) ≥ 1/3 + ǫ one has I(X) ≥ Cǫ.This Theorem is equivalent to Lemma 3.6 from [5].The key ingredient of the proof is the classification of minimal cycles (given by Lemmas 5.6, 5.7, 5.8 and Corollary 5.9).We do not use webs (as in [4], [5]) and operate with simplicial complexes.
Proof of Theorem 4.2 using Theorem A.1 and Theorem A.2. Let Cǫ be the constant given by Theorem A.2. Consider the set S of isomorphism types of all pure 2-complexes having at most 44 3 • C −2 ǫ faces.In particular, all complexes in S have at most 3 −1 • 44 3 • C −2 ǫ edges.Clearly, the set S is finite.We may present it as the disjoint union S = S1 ⊔ S2 where any S ∈ S1 satisfies ν(S) ≥ 1/3 + ǫ while for S ∈ S2 one has ν(S) < 1/3 + ǫ.By Theorem 2.1, a random complex XΓ contain as subcomplexes of X

Proof of Theorem A.2
Definition A.3. [12] We will 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 where Y ⊂ X is a proper tight subcomplex.Since ν(Y ) ≥ ν(X) for Y ⊂ X, it is obvious from (35) that it is enough to prove Theorem A.2 under the additional assumption that X is tight.
Remark A.4. Suppose that X is pure and tight and suppose that γ : S 1 → X is a simplicial loop with the ratio |γ| • AX(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) = AX (γ) and hence I(X ′ ) ≤ I(X) ≤ |γ| • AX(γ) −1 contradicting the assumption on γ.
Lemma A.6.Given ǫ > 0 there exists a constant C ′ ǫ > 0 such that for any finite pure tight connected 2-complex with ν(X) ≥ 1/3 + ǫ and L(X) ≤ 0 one has I(X) ≥ C ′ ǫ .This Lemma is similar to Theorem A.2 but it has an additional assumption that L(X) ≤ 0. It is clear from the proof that 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.We show that the number of isomorphism types of complexes X satisfying the conditions of the Lemma is finite; hence the statement of the Lemma follows by setting C ′ ǫ = min I(X) and using Corollary 5.9 which gives I(X) > 0 (since π1(X) is hyperbolic) and hence C ′ ǫ > 0. The inequality where e(X) denotes the number of 1-simplexes in X.By Lemma A.5 we have χ(X) = 1 − b1(X) ≤ 1 and using the assumption L(X) ≤ 0 we obtain e(X) ≤ ǫ −1 .This implies the finiteness of the set of possible isomorphism types of X and the result follows.
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|•AX (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 A.6.
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, ν(Y ) ≥ 1/3 + ǫ for any such Y .By Lemma A.5 we have that b2(X) = 0 which implies that b2(Y ) = 0. Besides, without loss of generality we may assume that Y is connected.The arguments of the proof of Lemma A.6 now apply (i.e.Y may have finitely many isomorphism types, each having a hyperbolic fundamental group) and it follows that minY I(Y ) ≥ C ′ ǫ where C ′ ǫ > 0 is a constant that only depends on ǫ.Hence if (36) holds we have I(X, X ′ ) ≥ minY I(Y ) ≥ C ′ ǫ and the result follows.Suppose that inequality (36) is false, i.e.I(X, X ′ ) < minY I(Y ), and consider a simplicial loop γ : S 1 → X ′ satisfying γ ∼ 1 in X and |γ| • AX (γ) −1 < minY 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 σ1, σ2 of D 2 with ψ(σ1) = ψ(σ2) the intersection σ1 ∩ σ2 is either ∅ or a vertex of D 2 .In other words, we exclude foldings, i.e. situations such that ψ(σ1) = ψ(σ2) and ∩ σ2 is an edge.Consider Z = X ′ ∪ ψ(D 2 ).Note that L(Z) ≤ 0. Indeed, since The main idea of the proof of Theorem A.2 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 A.7).The essential part of the proof is in estimating the area (the number of 2-simplexes) of such Σ.
Proof of Theorem A.2. Consider a connected tight pure 2-complex X satisfying ν(X) ≥ 1 3 + ǫ (37) and a simplicial prime loop γ : S 1 → X such that the ratio |γ| • AX (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 A.4, the map b is surjective.As explained in the proof of Lemma A.7, due to arguments of Ronan [25], we may assume that b has no foldings.For any integer i ≥ 1 we denote by Xi ⊂ X the pure subcomplex generated by all 2-simplexes σ of X such that the preimage b −1 (σ) ⊂ D 2 contains ≥ i two-dimensional simplexes.One has X = X1 ⊃ X2 ⊃ X3 ⊃ . . . .Each Xi may have several connected components and we will denote by Λ the set labelling all the connected components of the disjoint union ⊔ i≥1 Xi.For λ ∈ Λ the symbol X λ will denote the corresponding connected component of ⊔ i≥1 Xi and the symbol i = i(λ) ∈ {1, 2, . . .} will denote the index i ≥ 1 such that X λ is a connected component of Xi, viewed as a subset of ⊔ i≥1 Xi.We endow Λ with the following partial order: λ1 ≤ λ2 iff X λ 1 ⊃ X λ 2 (where X λ 1 and X λ 2 are viewed as subsets of X) and i(λ1) ≤ i(λ2).
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 (X) denotes the number of 2-simplexes in X.We have and Formula (39) follows from the observation that any 2-simplex of X = b(D 2 ) contributes to the RHS of (26) as many units as its multiplicity (the number of its preimages under b).Formula (40) follows from (39) 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 A.8.One has the following inequality See [12], Lemma 6.8 for the proof.Now we continue with the proof of Theorem A.2. Consider a tight pure 2-complex X satisfying (37) 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 Nj be the strongly connected components of N .Each Nj is PLhomeomorphic to a disc and we define ∂−Σ = ∪j∈J ∂Nj , the union of the circles ∂Nj 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(∂Nj) ⊂ X λ .Indeed, let λ1, . . ., λr ∈ Λ − be the minimal elements of Λ − with respect to the partial order introduced earlier.The complexes X λ 1 , . . ., 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 λ i , where i = 1, . . ., i. From our definition (38) it follows that the image of the circle b(∂Nj) is contained in the union ∪ r i=1 X λ i but since b(∂Nj ) is connected it must lie in one of the sets X λ i .We may apply Lemma A.7 to each of the circles ∂Nj .We obtain that each of the circles ∂Nj admits a spanning discs of area ≤ Kǫ|∂Nj |, where Kǫ = C ′−1 ǫ is the inverse of the constant given by Lemma A.7.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 ≤ Kǫ • |∂−Σ|.
This completes the proof of Theorem A.2.

2 .
Below we show that the set L has property (3) of Theorem 3.2.We start by describing examples of complexes from L. Example 1: Triangulated disc with one internal point and two added external edges (i.e.vi = 1 and c = 2).Example 2: Triangulated disc with two internal points and one added external edge (i.e.vi = 2 and c = 1).

Figure 2 :
Figure 2: Disc with 3 internal points as a quotient of a disc with no internal points; 3 pairs of adjacent edges are identified.

Figure 3 :
Figure 3: Example of a scroll without internal points.

Definition 5 . 3 .
A finite pure 2-complex Z is said to be a minimal cycle if b2(Z) = 1 and for any proper subcomplex Z ′ ⊂ Z one has b2(Z ′ ) = 0. Any minimal cycle is closed and strongly connected.Example 5.4.Let Z be the union of two subcomplexes Z = A ∪ B where each A and B is a triangulated projective plane and the intersection C = A ∩ B is a circle which is not null-homotopic in both A and B. Definition 5.5.A minimal cycle Z is said to be of type A if it has no proper closed 2-dimensional subcomplexes.If Z contains a proper closed 2-dimensional subcomplex then we say that Z is a minimal cycle of type B. Lemma 5.6.Let Z be a minimal cycle of type A satisfying ν(Z) > 1/3.Then Z is homotopy equivalent either to S 2 or to the wedge S 2 ∨ S 1 .Moreover, for any face σ ⊂ Z the boundary ∂σ is null-homotopic in Z − Int(σ).

Figure 6 :
Figure 6: Links of a vertex incident to an edge of degree 4.
subcomplex S ⊂ Y with e(S) ≤ 2ǫ −1 is collapsible to a graph.

7 2 -
torsion in fundamental groups of random clique complexes Theorem 7.1.Assume that