34 research outputs found
A language theoretic analysis of combings
A group is combable if it can be represented by a language of words
satisfying a fellow traveller property; an automatic group has a synchronous
combing which is a regular language. This paper gives a systematic analysis of
the properties of groups with combings in various formal language classes, and
of the closure properties of the associated classes of groups. It generalises
previous work, in particular of Epstein et al. and Bridson and Gilman.Comment: DVI and Post-Script files only, 21 pages. Submitted to International
Journal of Algebra and Computatio
Hairdressing in groups: a survey of combings and formal languages
A group is combable if it can be represented by a language of words
satisfying a fellow traveller property; an automatic group has a synchronous
combing which is a regular language. This article surveys results for combable
groups, in particular in the case where the combing is a formal language.Comment: 17 pages. Published copy, also available at
http://www.maths.warwick.ac.uk/gt/GTMon1/paper24.abs.htm
Ideal bicombings for hyperbolic groups and applications
For every hyperbolic group and more general hyperbolic graphs, we construct
an equivariant ideal bicombing: this is a homological analogue of the geodesic
flow on negatively curved manifolds. We then construct a cohomological
invariant which implies that several Measure Equivalence and Orbit Equivalence
rigidity results established by Monod-Shalom hold for all non-elementary
hyperbolic groups and their non-elementary subgroups. We also derive
superrigidity results for actions of general irreducible lattices on a large
class of hyperbolic metric spaces.Comment: Substantial generalizeation; now the results hold for a general class
of hyperbolic metric spaces (rather than just hyperbolic groups
Central extensions and bounded cohomology
It was shown by Gersten that a central extension of a finitely generated
group is quasi-isometrically trivial provided that its Euler class is bounded.
We say that a finitely generated group satisfies Property QITB
(quasi-isometrically trivial implies bounded) if the Euler class of any
quasi-isometrically trivial central extension of is bounded. We exhibit a
finitely generated group which does not satisfy Property QITB. This answers
a question by Neumann and Reeves, and provides partial answers to related
questions by Wienhard and Blank. We also prove that Property QITB holds for a
large class of groups, including amenable groups, right-angled Artin groups,
relatively hyperbolic groups with amenable peripheral subgroups, and 3-manifold
groups.
Finally, we show that Property QITB holds for every finitely presented group
if a conjecture by Gromov on bounded primitives of differential forms holds as
well.Comment: 32 pages. v2: Some proofs streamlined in view of recent work of
Milizia, revised connection to Gromov's conjectur