Large Random Simplicial Complexes, II; the fundamental groups

TL;DR

This paper investigates the fundamental groups of multi-parameter random simplicial complexes, establishing thresholds for properties like hyperbolicity, torsion, and asphericity, and analyzing their geometric and cohomological dimensions.

Contribution

It introduces a new class of random groups derived from simplicial complexes, providing thresholds and properties such as hyperbolicity, torsion, and asphericity.

Findings

Identified thresholds for hyperbolicity and nontrivial fundamental groups.

Found domains with 2-torsion and absence of odd-prime torsion.

Proved aspherical 2-dimensional subcomplexes satisfy the Whitehead Conjecture.

Abstract

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).