Collapse of random triangular groups: a closer look

TL;DR

This paper investigates the phase transition in random triangular groups, showing that they collapse when the number of relations exceeds a certain threshold proportional to n^{3/2}.

Contribution

It provides a precise threshold for collapse in random triangular groups, refining understanding of their asymptotic behavior.

Findings

Collapse occurs when t <= C n^{3/2} for some constant C.

The group remains infinite and hyperbolic for smaller t.

The threshold for collapse is sharper than previously known.

Abstract

The random triangular group \Gamma(n,t) is a group given by a presentation P=<S|R>, where S is a set of n generators and R is a random set of t cyclically reduced words of length three. The asymptotic behavior of \Gamma(n,t) is in some respects similar to that of widely studied density random group introduced by Gromov. In particular, it is known that if t <= n^{3/2-\epsilon} for some \epsilon > 0, then with probability 1-o(1) \Gamma(n,t) is infinite and hyperbolic, while for t >= n^{3/2+\epsilon}, with probability 1-o(1) it is trivial. In this note we show that \Gamma(n,t) collapses provided only that t <= C n^{3/2} for some constant C>0.