486 research outputs found
Finite -connected homogeneous graphs
A finite graph \G is said to be {\em -connected homogeneous}
if every isomorphism between any two isomorphic (connected) subgraphs of order
at most extends to an automorphism of the graph, where is a
group of automorphisms of the graph. In 1985, Cameron and Macpherson determined
all finite -homogeneous graphs. In this paper, we develop a method for
characterising -connected homogeneous graphs. It is shown that for a
finite -connected homogeneous graph \G=(V, E), either G_v^{\G(v)} is
--transitive or G_v^{\G(v)} is of rank and \G has girth , and
that the class of finite -connected homogeneous graphs is closed under
taking normal quotients. This leads us to study graphs where is
quasiprimitive on . We determine the possible quasiprimitive types for
in this case and give new constructions of examples for some possible types
Which finitely generated Abelian groups admit isomorphic Cayley graphs?
We show that Cayley graphs of finitely generated Abelian groups are rather
rigid. As a consequence we obtain that two finitely generated Abelian groups
admit isomorphic Cayley graphs if and only if they have the same rank and their
torsion parts have the same cardinality. The proof uses only elementary
arguments and is formulated in a geometric language.Comment: 16 pages; v2: added reference, reformulated quasi-convexity, v3:
small corrections; to appear in Geometriae Dedicat
Characterizing a vertex-transitive graph by a large ball
It is well-known that a complete Riemannian manifold M which is locally
isometric to a symmetric space is covered by a symmetric space. Here we prove
that a discrete version of this property (called local to global rigidity)
holds for a large class of vertex-transitive graphs, including Cayley graphs of
torsion-free lattices in simple Lie groups, and Cayley graph of torsion-free
virtually nilpotent groups. By contrast, we exhibit various examples of Cayley
graphs of finitely presented groups (e.g. SL(4,Z)) which fail to have this
property, answering a question of Benjamini, Ellis, and Georgakopoulos.
Answering a question of Cornulier, we also construct a continuum of non
pairwise isometric large-scale simply connected locally finite
vertex-transitive graphs. This question was motivated by the fact that
large-scale simply connected Cayley graphs are precisely Cayley graphs of
finitely presented groups and therefore have countably many isometric classes.Comment: v1: 38 pages. With an Appendix by Jean-Claude Sikorav v2: 48 pages.
Several improvements in the presentation. To appear in Journal of Topolog
Amenable hyperbolic groups
We give a complete characterization of the locally compact groups that are
non-elementary Gromov-hyperbolic and amenable. They coincide with the class of
mapping tori of discrete or continuous one-parameter groups of compacting
automorphisms. We moreover give a description of all Gromov-hyperbolic locally
compact groups with a cocompact amenable subgroup: modulo a compact normal
subgroup, these turn out to be either rank one simple Lie groups, or
automorphism groups of semi-regular trees acting doubly transitively on the set
of ends. As an application, we show that the class of hyperbolic locally
compact groups with a cusp-uniform non-uniform lattice, is very restricted.Comment: 41 pages, no figure. v2: revised version (minor changes
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