151 research outputs found
An exotic deformation of the hyperbolic space
On the one hand, we construct a continuous family of non-isometric proper
CAT(-1) spaces on which the isometry group of the
real hyperbolic -space acts minimally and cocompactly. This provides the
first examples of non-standard CAT(0) model spaces for simple Lie groups.
On the other hand, we classify all continuous non-elementary actions of on the infinite-dimensional real hyperbolic space. It
turns out that they are in correspondence with the exotic model spaces that we
construct.Comment: 42 pages, minor modifications, this is the final versio
Relative amenability
We introduce a relative fixed point property for subgroups of a locally
compact group, which we call relative amenability. It is a priori weaker than
amenability. We establish equivalent conditions, related among others to a
problem studied by Reiter in 1968. We record a solution to Reiter's problem.
We study the class X of groups in which relative amenability is equivalent to
amenability for all closed subgroups; we prove that X contains all familiar
groups. Actually, no group is known to lie outside X.
Since relative amenability is closed under Chabauty limits, it follows that
any Chabauty limit of amenable subgroups remains amenable if the ambient group
belongs to the vast class X.Comment: We added a solution to Reiter's problem and a discussion of
L^1-equivarianc
An indiscrete Bieberbach theorem: from amenable CAT(0) groups to Tits buildings
Non-positively curved spaces admitting a cocompact isometric action of an
amenable group are investigated. A classification is established under the
assumption that there is no global fixed point at infinity under the full
isometry group. The visual boundary is then a spherical building. When the
ambient space is geodesically complete, it must be a product of flats,
symmetric spaces, biregular trees and Bruhat--Tits buildings.
We provide moreover a sufficient condition for a spherical building arising
as the visual boundary of a proper CAT(0) space to be Moufang, and deduce that
an irreducible locally finite Euclidean building of dimension at least 2 is a
Bruhat--Tits building if and only if its automorphism group acts cocompactly
and chamber-transitively at infinity.Comment: minor typos corrected; reference adde
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
Decomposing locally compact groups into simple pieces
We present a contribution to the structure theory of locally compact groups. The emphasis is on compactly generated locally compact groups which admit no infinite discrete quotient. It is shown that such a group possesses a characteristic cocompact subgroup which is either connected or admits a non-compact non-discrete topologically simple quotient. We also provide a description of characteristically simple groups and of groups all of whose proper quotients are compact. We show that Noetherian locally compact groups without infinite discrete quotient admit a subnormal series with all subquotients compact, compactly generated Abelian, or compactly generated topologically simple. Two appendices introduce results and examples around the concept of quasi-produc
On the bounded cohomology of semi-simple groups, S-arithmetic groups and products
We prove vanishing results for Lie groups and algebraic groups (over any
local field) in bounded cohomology. The main result is a vanishing below twice
the rank for semi-simple groups. Related rigidity results are established for
S-arithmetic groups and groups over global fields. We also establish vanishing
and cohomological rigidity results for products of general locally compact
groups and their lattices
Some properties of non-positively curved lattices
We announce results on the structure of CAT(0) groups, CAT(0) lattices and of
the underlying spaces. Our statements rely notably on a general study of the
full isometry groups of proper CAT(0) spaces. Classical statements about
Hadamard manifolds are established for singular spaces; new arithmeticity and
rigidity statements are obtained.Comment: 6 page
Decomposing locally compact groups into simple pieces
We present a contribution to the structure theory of locally compact groups.
The emphasis is on compactly generated locally compact groups which admit no
infinite discrete quotient. It is shown that such a group possesses a
characteristic cocompact subgroup which is either connected or admits a
non-compact non-discrete topologically simple quotient. We also provide a
description of characteristically simple groups and of groups all of whose
proper quotients are compact. We show that Noetherian locally compact groups
without infinite discrete quotient admit a subnormal series with all
subquotients compact, compactly generated Abelian, or compactly generated
topologically simple. Two appendices introduce results and examples around the
concept of quasi-product.Comment: Index added; minor change
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