139 research outputs found
Hierarchically hyperbolic spaces I: curve complexes for cubical groups
In the context of CAT(0) cubical groups, we develop an analogue of the theory
of curve complexes and subsurface projections. The role of the subsurfaces is
played by a collection of convex subcomplexes called a \emph{factor system},
and the role of the curve graph is played by the \emph{contact graph}. There
are a number of close parallels between the contact graph and the curve graph,
including hyperbolicity, acylindricity of the action, the existence of
hierarchy paths, and a Masur--Minsky-style distance formula.
We then define a \emph{hierarchically hyperbolic space}; the class of such
spaces includes a wide class of cubical groups (including all virtually compact
special groups) as well as mapping class groups and Teichm\"{u}ller space with
any of the standard metrics. We deduce a number of results about these spaces,
all of which are new for cubical or mapping class groups, and most of which are
new for both. We show that the quasi-Lipschitz image from a ball in a nilpotent
Lie group into a hierarchically hyperbolic space lies close to a product of
hierarchy geodesics. We also prove a rank theorem for hierarchically hyperbolic
spaces; this generalizes results of Behrstock--Minsky, Eskin--Masur--Rafi,
Hamenst\"{a}dt, and Kleiner. We finally prove that each hierarchically
hyperbolic group admits an acylindrical action on a hyperbolic space. This
acylindricity result is new for cubical groups, in which case the hyperbolic
space admitting the action is the contact graph; in the case of the mapping
class group, this provides a new proof of a theorem of Bowditch.Comment: To appear in "Geometry and Topology". This version incorporates the
referee's comment
Arithmeticity vs. non-linearity for irreducible lattices
We establish an arithmeticity vs. non-linearity alternative for irreducible
lattices in suitable product groups, such as for instance products of
topologically simple groups. This applies notably to a (large class of)
Kac-Moody groups. The alternative relies on a CAT(0) superrigidity theorem, as
we follow Margulis' reduction of arithmeticity to superrigidity.Comment: 11 page
Separation of Relatively Quasiconvex Subgroups
Suppose that all hyperbolic groups are residually finite. The following
statements follow: In relatively hyperbolic groups with peripheral structures
consisting of finitely generated nilpotent subgroups, quasiconvex subgroups are
separable; Geometrically finite subgroups of non-uniform lattices in rank one
symmetric spaces are separable; Kleinian groups are subgroup separable. We also
show that LERF for finite volume hyperbolic 3-manifolds would follow from LERF
for closed hyperbolic 3-manifolds.
The method is to reduce, via combination and filling theorems, the
separability of a quasiconvex subgroup of a relatively hyperbolic group G to
the separability of a quasiconvex subgroup of a hyperbolic quotient G/N. A
result of Agol, Groves, and Manning is then applied.Comment: 22 pages, 2 figures. New version has numbering matching with the
published version in the Pacific Journal of Mathematics, 244 no. 2 (2010)
309--334
Systolic volume of homology classes
Given an integer homology class of a finitely presentable group, the systolic
volume quantifies how tight could be a geometric realization of this class. In
this paper, we study various aspects of this numerical invariant showing that
it is a complex and powerful tool to investigate topological properties of
homology classes of finitely presentable groups
A family of representations of braid groups on surfaces
We propose a family of new representations of the braid groups on surfaces
that extend linear representations of the braid groups on a disc such as the
Burau representation and the Lawrence-Krammer-Bigelow representation.Comment: 21 pages, 4 figure
Peripheral separability and cusps of arithmetic hyperbolic orbifolds
For X = R, C, or H it is well known that cusp cross-sections of finite volume
X-hyperbolic (n+1)-orbifolds are flat n-orbifolds or almost flat orbifolds
modelled on the (2n+1)-dimensional Heisenberg group N_{2n+1} or the
(4n+3)-dimensional quaternionic Heisenberg group N_{4n+3}(H). We give a
necessary and sufficient condition for such manifolds to be diffeomorphic to a
cusp cross-section of an arithmetic X-hyperbolic (n+1)-orbifold. A principal
tool in the proof of this classification theorem is a subgroup separability
result which may be of independent interest.Comment: Published by Algebraic and Geometric Topology at
http://www.maths.warwick.ac.uk/agt/AGTVol4/agt-4-32.abs.htm
Frobenius groups of automorphisms and their fixed points
Suppose that a finite group admits a Frobenius group of automorphisms
with kernel and complement such that the fixed-point subgroup of
is trivial: . In this situation various properties of are
shown to be close to the corresponding properties of . By using
Clifford's theorem it is proved that the order is bounded in terms of
and , the rank of is bounded in terms of and the rank
of , and that is nilpotent if is nilpotent. Lie ring
methods are used for bounding the exponent and the nilpotency class of in
the case of metacyclic . The exponent of is bounded in terms of
and the exponent of by using Lazard's Lie algebra associated with the
Jennings--Zassenhaus filtration and its connection with powerful subgroups. The
nilpotency class of is bounded in terms of and the nilpotency class
of by considering Lie rings with a finite cyclic grading satisfying a
certain `selective nilpotency' condition. The latter technique also yields
similar results bounding the nilpotency class of Lie rings and algebras with a
metacyclic Frobenius group of automorphisms, with corollaries for connected Lie
groups and torsion-free locally nilpotent groups with such groups of
automorphisms. Examples show that such nilpotency results are no longer true
for non-metacyclic Frobenius groups of automorphisms.Comment: 31 page
The space of Anosov diffeomorphisms
We consider the space \X of Anosov diffeomorphisms homotopic to a fixed
automorphism of an infranilmanifold . We show that if is the 2-torus
then \X is homotopy equivalent to . In contrast,
if dimension of is large enough, we show that \X is rich in homotopy and
has infinitely many connected components.Comment: Version 2: referee suggestions result in a better expositio
Quadratic equations over free groups are NP-complete
We prove that the problems of deciding whether a quadratic equation over a
free group has a solution is NP-complete
Internal structures in n-permutable varieties
We analyze the notions of reflexive multiplicative graph, internal category and internal groupoid for n-permutable varieties. (C) 2012 Elsevier B.V. All rights reserved.CMUC; FCT (Portugal) through European Program COMPETE/FEDERinfo:eu-repo/semantics/publishedVersio
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