Abstract

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.

Highlights

  • A clique in a graph Γ is a set of vertices of Γ such that any two of them are connected by an edge

  • 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

  • For a random graph Γ ∈ G(n, p) the 2-skeleton XΓ(2) of the clique complex XΓ has the following property with probability tending to 1 as n → ∞: a subcomplex Y ⊂ XΓ(2) is aspherical if and only if every subcomplex S ⊂ Y having at most 2ǫ−1 edges is aspherical

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Summary

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 following theorem states that 2-torsion appears in the fundamental group of a random clique complex when we cross the threshold 11/30: Theorem C: [See Theorem 7.2] Assume that n−11/30 ≪ p ≪ n−1/3−ǫ (5). 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. 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 ⊂ XΓ(2) satisfies the Whitehead Conjecture, i.e. any subcomplex Y ′ ⊂ Y is aspherical. The authors thank the referee for making useful critical remarks

The containment problem
Threshold for collapsibility to a graph
Uniform hyperbolicity
The Whitehead Conjecture
The number of combinatorial embeddings
Projective planes in clique complexes of random graphs
Absence of odd torsion
Findings
A Appendix
Full Text
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