1,709 research outputs found
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
Characterisations of algebraic properties of groups in terms of harmonic functions
We prove various results connecting structural or algebraic properties of
graphs and groups to conditions on their spaces of harmonic functions. In
particular: we show that a group with a finitely supported symmetric measure
has a finite-dimensional space of harmonic functions if and only if it is
virtually cyclic; we present a new proof of a result of V. Trofimov that an
infinite vertex-transitive graph admits a non-constant harmonic function; we
give a new proof of a result of T. Ceccherini-Silberstein, M. Coornaert and J.
Dodziuk that the Laplacian on an infinite, connected, locally finite graph is
surjective; and we show that the positive harmonic functions on a non-virtually
nilpotent linear group span an infinite-dimensional space.Comment: 32 pages, 2 figures. Final version. Most of the material on virtually
nilpotent groups from V1 is now superseded by http://arxiv.org/abs/1505.0117
On solvable minimally transitive permutation groups
We investigate properties of finite transitive permutation groups in which all proper subgroups of act intransitively on
In particular, we are interested in reduction theorems for minimally transitive
representations of solvable groups.Comment: 8 pages, no figures, journal pape
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