Outer actions of Out(Fn) on small right-angled Artin groups

Kielak, Dawid

doi:10.2140/agt.2018.18.1041

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Outer actions of $\mathrm{Out}(F_n)$ on small right-angled Artin groups

Dawid Kielak

Algebraic & Geometric Topology 18 (2018) 1041–1065

DOI: 10.2140/agt.2018.18.1041

Abstract

We determine the precise conditions under which $SOut (F_{n})$ , the unique index-two subgroup of $Out (F_{n})$ , can act nontrivially via outer automorphisms on a RAAG whose defining graph has fewer than $\frac{1}{2} (\binom{n}{2})$ vertices.

We also show that the outer automorphism group of a RAAG cannot act faithfully via outer automorphisms on a RAAG with a strictly smaller (in number of vertices) defining graph.

Along the way we determine the minimal dimensions of nontrivial linear representations of congruence quotients of the integral special linear groups over algebraically closed fields of characteristic zero, and provide a new lower bound on the cardinality of a set on which $SOut (F_{n})$ can act nontrivially.

Keywords

Out(F_n), RAAGs

Mathematical Subject Classification 2010

Primary: 20F65

Secondary: 20F28, 20F36

References

Publication

Received: 25 February 2017

Revised: 14 August 2017

Accepted: 21 November 2017

Published: 12 March 2018

Authors

Dawid Kielak
Fakultät für Mathematik Universität Bielefeld Bielefeld Germany

Volume 18, issue 2 (2018)