## Detecting topological properties of boundaries of hyperbolic groups

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##### Authors

Barrett, Benjamin James

##### Advisors

Wilton, Henry John Rutley

##### Date

2018-11-24##### Awarding Institution

University of Cambridge

##### Author Affiliation

Pure Mathematics and Mathematical Statistics

##### Qualification

Doctor of Philosophy (PhD)

##### Language

English

##### Type

Thesis

##### Metadata

Show full item record##### Citation

Barrett, B. J. (2018). Detecting topological properties of boundaries of hyperbolic groups (Doctoral thesis). https://doi.org/10.17863/CAM.32926

##### Abstract

In general, a finitely presented group can have very nasty properties, but many of these
properties are avoided if the group is assumed to admit a nice action by isometries on a
space with a negative curvature property, such as Gromov hyperbolicity. Such groups are
surprisingly common: there is a sense in which a random group admits such an action, as
do some groups of classical interest, such as fundamental groups of closed Riemannian
manifolds with negative sectional curvature. If a group admits an action on a Gromov
hyperbolic space then large scale properties of the space give useful invariants of the group.
One particularly natural large scale property used in this way is the Gromov boundary.
The Gromov boundary of a hyperbolic group is a compact metric space that is, in a
sense, approximated by spheres of large radius in the Cayley graph of the group. The
technical results contained in this thesis are effective versions of this statement: we see
that the presence of a particular topological feature in the boundary of a hyperbolic group
is determined by the geometry of balls in the Cayley graph of radius bounded above by
some known upper bound, and is therefore algorithmically detectable.
Using these technical results one can prove that certain properties of a group can
be computed from its presentation. In particular, we show that there are algorithms
that, when given a presentation for a one-ended hyperbolic group, compute Bowditch’s
canonical decomposition of that group and determine whether or not that group is
virtually Fuchsian.
The final chapter of this thesis studies the problem of detecting Cech cohomological
features in boundaries of hyperbolic groups. Epstein asked whether there is an algorithm
that computes the Cech cohomology of the boundary of a given hyperbolic group. We
answer Epstein’s question in the affirmative for a restricted class of hyperbolic groups:
those that are fundamental groups of graphs of free groups with cyclic edge groups. We also
prove the computability of the Cech cohomology of a space with some similar properties
to the boundary of a hyperbolic group: Otal’s decomposition space associated to a line
pattern in a free group.

##### Keywords

Geometry, Geometric group theory, Hyperbolic groups, JSJ theory

##### Sponsorship

EPSRC studentship

##### Identifiers

This record's DOI: https://doi.org/10.17863/CAM.32926

##### Rights

Attribution 4.0 International (CC BY 4.0)

Licence URL: https://creativecommons.org/licenses/by/4.0/

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