77,355 research outputs found
Percolation on nonunimodular transitive graphs
We extend some of the fundamental results about percolation on unimodular
nonamenable graphs to nonunimodular graphs. We show that they cannot have
infinitely many infinite clusters at critical Bernoulli percolation. In the
case of heavy clusters, this result has already been established, but it also
follows from one of our results. We give a general necessary condition for
nonunimodular graphs to have a phase with infinitely many heavy clusters. We
present an invariant spanning tree with on some nonunimodular graph.
Such trees cannot exist for nonamenable unimodular graphs. We show a new way of
constructing nonunimodular graphs that have properties more peculiar than the
ones previously known.Comment: Published at http://dx.doi.org/10.1214/009117906000000494 in the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Finite and infinite quotients of discrete and indiscrete groups
These notes are devoted to lattices in products of trees and related topics.
They provide an introduction to the construction, by M. Burger and S. Mozes, of
examples of such lattices that are simple as abstract groups. Two features of
that construction are emphasized: the relevance of non-discrete locally compact
groups, and the two-step strategy in the proof of simplicity, addressing
separately, and with completely different methods, the existence of finite and
infinite quotients. A brief history of the quest for finitely generated and
finitely presented infinite simple groups is also sketched. A comparison with
Margulis' proof of Kneser's simplicity conjecture is discussed, and the
relevance of the Classification of the Finite Simple Groups is pointed out. A
final chapter is devoted to finite and infinite quotients of hyperbolic groups
and their relation to the asymptotic properties of the finite simple groups.
Numerous open problems are discussed along the way.Comment: Revised according to referee's report; definition of BMW-groups
updated; more examples added in Section 4; new Proposition 5.1
Separable and tree-like asymptotic cones of groups
Using methods from nonstandard analysis, we will discuss which metric spaces
can be realized as asymptotic cones. Applying the results we will find in the
context of groups, we will prove that a group with "a few" separable asymptotic
cones is virtually nilpotent, and we will classify the real trees appearing as
asymptotic cones of (not necessarily hyperbolic) groups.Comment: The hypothesis of Theorem 1.2 had to be strengthene
JSJ-decompositions of finitely presented groups and complexes of groups
A JSJ-splitting of a group over a certain class of subgroups is a graph
of groups decomposition of which describes all possible decompositions of
as an amalgamated product or an HNN extension over subgroups lying in the
given class. Such decompositions originated in 3-manifold topology. In this
paper we generalize the JSJ-splitting constructions of Sela, Rips-Sela and
Dunwoody-Sageev and we construct a JSJ-splitting for any finitely presented
group with respect to the class of all slender subgroups along which the group
splits. Our approach relies on Haefliger's theory of group actions on CAT
spaces
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