101 research outputs found
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
Finiteness properties of soluble arithmetic groups over global function fields
Let G be a Chevalley group scheme and B<=G a Borel subgroup scheme, both
defined over Z. Let K be a global function field, S be a finite non-empty set
of places over K, and O_S be the corresponding S-arithmetic ring. Then, the
S-arithmetic group B(O_S) is of type F_{|S|-1} but not of type FP_{|S|}.
Moreover one can derive lower and upper bounds for the geometric invariants
\Sigma^m(B(O_S)). These are sharp if G has rank 1. For higher ranks, the
estimates imply that normal subgroups of B(O_S) with abelian quotients,
generically, satisfy strong finiteness conditions.Comment: Published by Geometry and Topology at
http://www.maths.warwick.ac.uk/gt/GTVol8/paper15.abs.htm
Moufang sets of finite Morley rank of odd type
We show that for a wide class of groups of finite Morley rank the presence of
a split -pair of Tits rank forces the group to be of the form
and the -pair to be standard. Our approach is via
the theory of Moufang sets. Specifically, we investigate infinite and so-called
hereditarily proper Moufang sets of finite Morley rank in the case where the
little projective group has no infinite elementary abelian -subgroups and
show that all such Moufang sets are standard (and thus associated to
for an algebraically closed field of
characteristic not ) provided the Hua subgroups are nilpotent. Further, we
prove that the same conclusion can be reached whenever the Hua subgroups are
-groups and the root groups are not simple
On epimorphisms of spherical Moufang buildings
In this paper we classify the the epimorphisms of irreducible spherical
Moufang buildings (of rank at least 2) defined over a field. As an application
we characterize indecomposable epimorphisms of these buildings as those
epimorphisms arising from R-buildings.Comment: revised version, 44 page
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