45 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
Low dimensional free and linear representations of
We study homomorphisms from to , and
for , where is a field of characteristic other
than 2 or 3. We conclude that all -linear representations of dimension at
most 6 of factor through , and that all
homomorphisms from to have finite
image.Comment: Final versio
Nielsen Realisation by Gluing: Limit Groups and Free Products
We generalise the Karrass-Pietrowski-Solitar and the Nielsen realisation
theorems from the setting of free groups to that of free products. As a result,
we obtain a fixed point theorem for finite groups of outer automorphisms acting
on the relative free splitting complex of Handel--Mosher and on the outer space
of a free product of Guirardel--Levitt, as well as a relative version of the
Nielsen realisation theorem, which in the case of free groups answers a
question of Karen Vogtmann. We also prove Nielsen realisation for limit groups,
and as a byproduct obtain a new proof that limit groups are CAT(). The
proofs rely on a new version of Stallings' theorem on groups with at least two
ends, in which some control over the behaviour of virtual free factors is
gained.Comment: 28 pages, 1 figur
The 6-strand braid group is CAT(0)
We show that braid groups with at most 6 strands are CAT(0) using the close
connection between these groups, the associated non-crossing partition
complexes and the embeddability of their diagonal links into spherical
buildings of type A. Furthermore, we prove that the orthoscheme complex of any
bounded graded modular complemented lattice is CAT(0), giving a partial answer
to a conjecture of Brady and McCammond.Comment: 27 pages, 13 figures. To appear in Geometriae Dedicata, the final
publication is available at Springer via
http://dx.doi.org/10.1007/s10711-015-0138-