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

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.

29 pages. Mostly rewritten, results tightened, statements corrected, gaps filled

References in corpus (1)

• Random groups arising as graph products

Cited by in corpus (4)

• Quasi-isometric classification of right-angled Artin groups I: the finite out case
• Relative automorphism groups of right-angled Artin groups
• Groups quasi-isometric to RAAG's
• Trees, homology, and automorphism groups of RAAGs