101 research outputs found

    An indiscrete Bieberbach theorem: from amenable CAT(0) groups to Tits buildings

    Get PDF
    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

    Full text link
    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

    Full text link
    We show that for a wide class of groups of finite Morley rank the presence of a split BNBN-pair of Tits rank 11 forces the group to be of the form PSL2\operatorname{PSL}_2 and the BNBN-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 22-subgroups and show that all such Moufang sets are standard (and thus associated to PSL2(F)\operatorname{PSL}_2(F) for FF an algebraically closed field of characteristic not 22) provided the Hua subgroups are nilpotent. Further, we prove that the same conclusion can be reached whenever the Hua subgroups are LL-groups and the root groups are not simple

    On epimorphisms of spherical Moufang buildings

    Get PDF
    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
    corecore