199 research outputs found
Kazhdan and Haagerup Properties in algebraic groups over local fields
Given a Lie algebra \s, we call Lie \s-algebra a Lie algebra endowed with a
reductive action of \s. We characterize the minimal \s-Lie algebras with a
nontrivial action of \s, in terms of irreducible representations of \s and
invariant alternating forms.
As a first application, we show that if \g is a Lie algebra over a field of
characteristic zero whose amenable radical is not a direct factor, then \g
contains a subalgebra which is isomorphic to the semidirect product of sl_2 by
either a nontrivial irreducible representation or a Heisenberg group (this was
essentially due to Cowling, Dorofaeff, Seeger, and Wright). As a corollary, if
G is an algebraic group over a local field K of characteristic zero, and if its
amenable radical is not, up to isogeny, a direct factor, then G(K) has Property
(T) relative to a noncompact subgroup. In particular, G(K) does not have
Haagerup's property. This extends a similar result of Cherix, Cowling and
Valette for connected Lie groups, to which our method also applies.
We give some other applications. We provide a characterization of connected
Lie groups all of whose countable subgroups have Haagerup's property. We give
an example of an arithmetic lattice in a connected Lie group which does not
have Haagerup's property, but has no infinite subgroup with relative Property
(T). We also give a continuous family of pairwise non-isomorphic connected Lie
groups with Property (T), with pairwise non-isomorphic (resp. isomorphic) Lie
algebras.Comment: 11 pages, no figur
Relative Kazhdan Property
We perform a systematic investigation of Kazhdan's relative Property (T) for
pairs (G,X), where G a locally compact group and X is any subset. When G is a
connected Lie group or a p-adic algebraic group, we provide an explicit
characterization of subsets X of G such that (G,X) has relative Property (T).
In order to extend this characterization to lattices of G, a notion of
"resolutions" is introduced, and various characterizations of it are given.
Special attention is paid to subgroups of SU(2,1) and SO(4,1).Comment: 36 pages, no figure; to appear in Ann. Sci. Ecole Norm. Su
Infinite groups with large balls of torsion elements and small entropy
We exhibit infinite, solvable, virtually abelian groups with a fixed number
of generators, having arbitrarily large balls consisting of torsion elements.
We also provide a sequence of 3-generator non-virtually nilpotent polycyclic
groups of algebraic entropy tending to zero. All these examples are obtained by
taking appropriate quotients of finitely presented groups mapping onto the
first Grigorchuk group.Comment: 8 pages; to appear in Archiv der Mathemati
On conjugacy growth for solvable groups
We prove that a finitely generated solvable group which is not virtually
nilpotent has exponential conjugacy growth.Comment: 7 pages, no figur
On groups with weak Sierpi\'nski subsets
In a group , a weak Sierpi\'nski subset is a subset such that for some
and , we have and
. In this setting, we study the subgroup generated by
and , and show that it has a special presentation, namely of the form
unless it is free over . In
addition, in such groups , we characterize all weak Sierpi\'nski subsets
Homological finiteness properties of wreath products
We study the homological finiteness property FPm of permutational wreath
products
Isometric group actions on Hilbert spaces: growth of cocycles
We study growth of 1-cocycles of locally compact groups, with values in
unitary representations. Discussing the existence of 1-cocycles with linear
growth, we obtain the following alternative for a class of amenable groups G
containing polycyclic groups and connected amenable Lie groups: either G has no
quasi-isometric embedding into Hilbert space, or G admits a proper cocompact
action on some Euclidean space.
On the other hand, noting that almost coboundaries (i.e. 1-cocycles
approximable by bounded 1-cocycles) have sublinear growth, we discuss the
converse, which turns out to hold for amenable groups with "controlled" Folner
sequences; for general amenable groups we prove the weaker result that
1-cocycles with sufficiently small growth are almost coboundaries. Besides, we
show that there exist, on a-T-menable groups, proper cocycles with arbitrary
small growth.Comment: 26 pages, no figure. To appear in Geom. Funct. Ana
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