780 research outputs found
The random graph
Erd\H{o}s and R\'{e}nyi showed the paradoxical result that there is a unique
(and highly symmetric) countably infinite random graph. This graph, and its
automorphism group, form the subject of the present survey.Comment: Revised chapter for new edition of book "The Mathematics of Paul
Erd\H{o}s
Discrete isometry groups of symmetric spaces
This survey is based on a series of lectures that we gave at MSRI in Spring
2015 and on a series of papers, mostly written jointly with Joan Porti. Our
goal here is to:
1. Describe a class of discrete subgroups of higher rank
semisimple Lie groups, which exhibit some "rank 1 behavior".
2. Give different characterizations of the subclass of Anosov subgroups,
which generalize convex-cocompact subgroups of rank 1 Lie groups, in terms of
various equivalent dynamical and geometric properties (such as asymptotically
embedded, RCA, Morse, URU).
3. Discuss the topological dynamics of discrete subgroups on flag
manifolds associated to and Finsler compactifications of associated
symmetric spaces . Find domains of proper discontinuity and use them to
construct natural bordifications and compactifications of the locally symmetric
spaces .Comment: 77 page
On covering by translates of a set
In this paper we study the minimal number of translates of an arbitrary
subset of a group needed to cover the group, and related notions of the
efficiency of such coverings. We focus mainly on finite subsets in discrete
groups, reviewing the classical results in this area, and generalizing them to
a much broader context. For example, we show that while the worst-case
efficiency when has elements is of order , for fixed and
large, almost every -subset of any given -element group covers
with close to optimal efficiency.Comment: 41 pages; minor corrections; to appear in Random Structures and
Algorithm
- …