Finiteness of outer automorphism groups of random right-angled Artin groups

TL;DR

This paper determines the precise edge probability thresholds in random graphs that almost surely result in finite or infinite outer automorphism groups of the associated right-angled Artin groups, refining previous bounds.

Contribution

It establishes exact asymptotic bounds for when Out(A_Gamma) is finite or infinite in random right-angled Artin groups, sharpening earlier results.

Findings

Out(A_Gamma) is almost surely finite within specific edge probability bounds.

Out(A_Gamma) is almost surely infinite outside these bounds.

Provides a sharp threshold for the finiteness of automorphism groups in random graph models.

Abstract

We consider the outer automorphism group Out(A_Gamma) of the right-angled Artin group A_Gamma of a random graph Gamma on n vertices in the Erdos--Renyi model. We show that the functions (log(n)+log(log(n)))/n and 1-(log(n)+log(log(n)))/n bound the range of edge probability functions for which Out(A_Gamma) is finite: if the probability of an edge in Gamma is strictly between these functions as n grows, then asymptotically Out(A_Gamma) is almost surely finite, and if the edge probability is strictly outside of both of these functions, then asymptotically Out(A_Gamma) is almost surely infinite. This sharpens results of Ruth Charney and Michael Farber from their preprint "Random groups arising as graph products", arXiv:1006.3378v1.