## Algebraic & Geometric Topology

### Finiteness of outer automorphism groups of random right-angled {A}rtin groups

Matthew B Day

#### Abstract

We consider the outer automorphism group $Out(AΓ)$ of the right-angled Artin group $AΓ$ of a random graph $Γ$ on $n$ vertices in the Erdős–Rényi model. We show that the functions $n−1(log(n)+ log(log(n)))$ and $1−n−1(log(n)+ log(log(n)))$ bound the range of edge probability functions for which $Out(AΓ)$ is finite: if the probability of an edge in $Γ$ is strictly between these functions as $n$ grows, then asymptotically $Out(AΓ)$ is almost surely finite, and if the edge probability is strictly outside of both of these functions, then asymptotically $Out(AΓ)$ is almost surely infinite. This sharpens a result of Ruth Charney and Michael Farber.

#### Article information

Source
Algebr. Geom. Topol., Volume 12, Number 3 (2012), 1553-1583.

Dates
Revised: 10 April 2012
Accepted: 25 April 2012
First available in Project Euclid: 19 December 2017

https://projecteuclid.org/euclid.agt/1513715407

Digital Object Identifier
doi:10.2140/agt.2012.12.1553

Mathematical Reviews number (MathSciNet)
MR2966695

Zentralblatt MATH identifier
1246.05141

#### Citation

Day, Matthew B. Finiteness of outer automorphism groups of random right-angled {A}rtin groups. Algebr. Geom. Topol. 12 (2012), no. 3, 1553--1583. doi:10.2140/agt.2012.12.1553. https://projecteuclid.org/euclid.agt/1513715407

#### References

• B Bollobás, Random graphs, Academic Press, London (1985)
• R Charney, M Farber, Random groups arising as graph products, to appear in Alg. Geom. Topol. 12 (2012) 979–995
• A Costa, M Farber, Topology of random right angled Artin groups, J. Topol. Anal. 3 (2011) 69–87
• P Erdős, A Rényi, On random graphs I, Publ. Math. Debrecen 6 (1959) 290–297
• M R Laurence, A generating set for the automorphism group of a graph group, J. London Math. Soc. 52 (1995) 318–334
• W Rudin, Real and complex analysis, third edition, McGraw–Hill, New York (1987)