835,694 research outputs found
Regular dessins with a given automorphism group
Dessins d'enfants are combinatorial structures on compact Riemann surfaces
defined over algebraic number fields, and regular dessins are the most
symmetric of them. If G is a finite group, there are only finitely many regular
dessins with automorphism group G. It is shown how to enumerate them, how to
represent them all as quotients of a single regular dessin U(G), and how
certain hypermap operations act on them. For example, if G is a cyclic group of
order n then U(G) is a map on the Fermat curve of degree n and genus
(n-1)(n-2)/2. On the other hand, if G=A_5 then U(G) has genus
274218830047232000000000000000001. For other non-abelian finite simple groups,
the genus is much larger.Comment: 19 page
Polygonal Complexes and Graphs for Crystallographic Groups
The paper surveys highlights of the ongoing program to classify discrete
polyhedral structures in Euclidean 3-space by distinguished transitivity
properties of their symmetry groups, focussing in particular on various aspects
of the classification of regular polygonal complexes, chiral polyhedra, and
more generally, two-orbit polyhedra.Comment: 21 pages; In: Symmetry and Rigidity, (eds. R.Connelly, A.Ivic Weiss
and W.Whiteley), Fields Institute Communications, to appea
On groups with Property (T_lp)
Let p be a real number with 1<p and different from 2. We study Property
(T_lp) for a second countable locally compact group G. Property (T_lp) is a
weak version of Kazhdan's Property (T), defined in terms of the orthogonal
representations of G on the sequence space lp. We show that Property (T_lp) for
a totally disconnected group G is characterized by an isolation property of the
trivial representation from the quasi-regular representations associated to
open subgroups of G. Groups with Property (T_lp) share some important
properties with Kazhdan groups (compact generation, compact abelianization,
...). Simple algebraic groups over non-archimedean local fields as well as
automorphism groups of regular trees have Property (T_lp). In the case of
discrete groups, Property (T_lp) implies Lubotzky's Property tau and is implied
by Property (F) of Glasner and Monod. We show that an irreducible lattice in a
product of two locally compact groups G and H have Property (T_lp), whenever G
has Property (T) and H is connected and minimally almost periodic.Comment: 17 page
A unified proof of the Howe-Moore property
We provide a unified proof of all known examples of locally compact groups
that enjoy the Howe-Moore property, namely, the vanishing at infinity of all
matrix coefficients of the group unitary representations that are without
non-zero invariant vectors. These examples are: connected, non-compact, simple
real Lie groups with finite center, isotropic simple algebraic groups over non
Archimedean local fields and closed, topologically simple subgroups of Aut(T)
that act 2-transitively on the boundary at infinity of T, where T is a
bi-regular tree with valence > 2 at every vertex.Comment: Final version, to appear in Journal of Lie Theor
- …