30 research outputs found
Outer actions of on small right-angled Artin groups
We determine the precise conditions under which , the
unique index two subgroup of , can act non-trivially via
outer automorphisms on a RAAG whose defining graph has fewer than 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 non-trivial 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 can act
non-trivially.Comment: 16 pages v.2 Minor changes. Final versio
On the number of outer automorphisms of the automorphism group of a right-angled Artin group
We show that there is no uniform upper bound on |Out(Aut(A))| when A ranges
over all right-angled Artin groups. This is in contrast with the cases where A
is free or free abelian: for all n, Dyer-Formanek and Bridson-Vogtmann showed
that Out(Aut(F_n)) = 1, while Hua-Reiner showed |Out(Aut(Z^n)| = |Out(GL(n,Z))|
< 5. We also prove the analogous theorem for Out(Out(A)). We establish our
results by giving explicit examples; one useful tool is a new class of graphs
called austere graphs
Calculating the virtual cohomological dimension of the automorphism group of a RAAG
We describe an algorithm to find the virtual cohomological dimension of the
automorphism group of a right-angled Artin group. The algorithm works in the
relative setting; in particular it also applies to untwisted automorphism
groups and basis-conjugating automorphism groups. The main new tool is the
construction of free abelian subgroups of certain Fouxe-Rabinovitch groups of
rank equal to their virtual cohomological dimension, generalizing a result of
Meucci in the setting of free groups.Comment: 15 pages, 2 figures. Revised background on RORGs, small changes
elsewhere. Accepted to appear in Bulletin of the LM