4,845 research outputs found
Geometry of generated groups with metrics induced by their Cayley color graphs
Let be a group and let be a generating set of . In this article,
we introduce a metric on with respect to , called the cardinal
metric. We then compare geometric structures of and ,
where denotes the word metric. In particular, we prove that if is
finite, then and are not quasi-isometric in the case when
has infinite diameter and they are bi-Lipschitz equivalent
otherwise. We also give an alternative description of cardinal metrics by using
Cayley color graphs. It turns out that color-permuting and color-preserving
automorphisms of Cayley digraphs are isometries with respect to cardinal
metrics
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
Automorphisms of graph products of groups from a geometric perspective
This article studies automorphism groups of graph products of arbitrary
groups. We completely characterise automorphisms that preserve the set of
conjugacy classes of vertex groups as those automorphisms that can be
decomposed as a product of certain elementary automorphisms (inner
automorphisms, partial conjugations, automorphisms associated to symmetries of
the underlying graph). This allows us to completely compute the automorphism
group of certain graph products, for instance in the case where the underlying
graph is finite, connected, leafless and of girth at least . If in addition
the underlying graph does not contain separating stars, we can understand the
geometry of the automorphism groups of such graph products of groups further:
we show that such automorphism groups do not satisfy Kazhdan's property (T) and
are acylindrically hyperbolic. Applications to automorphism groups of graph
products of finite groups are also included. The approach in this article is
geometric and relies on the action of graph products of groups on certain
complexes with a particularly rich combinatorial geometry. The first such
complex is a particular Cayley graph of the graph product that has a
quasi-median geometry, a combinatorial geometry reminiscent of (but more
general than) CAT(0) cube complexes. The second (strongly related) complex used
is the Davis complex of the graph product, a CAT(0) cube complex that also has
a structure of right-angled building.Comment: 36 pages. The article subsumes and vastly generalises our preprint
arXiv:1803.07536. To appear in Proc. Lond. Math. So
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