Fullarton, Neil James
(2014)
Palindromic automorphisms of free groups and rigidity of automorphism groups of rightangled Artin groups.
PhD thesis, University of Glasgow.
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Abstract
Let F_n denote the free group of rank n with free basis X. The palindromic automorphism group PiA_n of F_n consists of automorphisms taking each member of X to a palindrome: that is, a word on X that reads the same backwards as forwards. We obtain finite generating sets for certain stabiliser subgroups of PiA_n. We use these generating sets to find an infinite generating set for the socalled palindromic Torelli group PI_n, the subgroup of PiA_n consisting of palindromic automorphisms inducing the identity on the
abelianisation of F_n. Two crucial tools for finding this generating set are a new simplicial complex, the socalled complex of partial pibases, on which PiA_n acts, and a Birman exact sequence for PiA_n, which allows us to induct on n.
We also obtain a rigidity result for automorphism groups of rightangled Artin groups. Let G be a finite simplicial graph, defining the rightangled Artin group A_G. We show that as A_G ranges over all rightangled Artin groups, the order of Out(Aut(A_G)) does not have a uniform upper bound. This is in contrast with extremal cases when A_G is free or free abelian: in these cases, Out(Aut(A_G)) < 5. We prove that no uniform upper bound exists in general by placing constraints on the graph G that yield tractable decompositions of Aut(A_G). These decompositions allow us to construct explicit members of Out(Aut(A_G)).
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