Location of Repository

Conjugacy and subgroups of word-hyperbolic groups

By David John Buckley


This thesis describes a number of algorithms and properties relating to Gromov’s\ud word-hyperbolic groups. A fuller outline of the thesis is given, and a number of\ud basic concepts relating to metric spaces, hyperbolicity and automaticity are first\ud briefly detailed in Chapter 1. Chapter 2 then details a solution to the conjugacy\ud problem for lists of elements in a word-hyperbolic group which can be run in linear\ud time; this is an improvement on a quadratic time algorithm for lists which contain\ud an infinite order element. Chapter 3 provides a number of further results and\ud algorithms which build upon this result to efficiently solve problems relating to quasiconvex\ud subgroups of word-hyperbolic groups – specifically, the problem of testing\ud if an element conjugates into a quasiconvex subgroup, and testing equality of double\ud cosets. In Chapter 4, a number of properties of certain coset Cayley graphs are\ud studied, in particular showing that graph morphisms which preserve edge labels and\ud directions and map a quasiconvex subset to a single point also preserve a variety of\ud other properties, for instance hyperbolicity. Finally, Chapter 5 gives a proof that all\ud word-hyperbolic groups are 14-hyperbolic with respect to some generating set

Topics: QA
OAI identifier: oai:wrap.warwick.ac.uk:3631

Suggested articles



  1. (1995). A Non-quasiconvex Subgroup of a Hyperbolic Group with an Exotic Limit Set.
  2. (1994). Detecting quasiconvexity: algorithmic aspects. Geometric and computational perspectives on infinite groups (Minneapolis,
  3. (2009). Diophantine geometry over groups VII: the elementary theory of a hyperbolic group. doi
  4. (1987). Hyperbolic groups. Essays in group theory, doi
  5. (1979). Introduction to Automata Theory, Languages, and Computation. doi
  6. (1991). Rational subgroups of biautomatic groups. doi
  7. (1976). Real-time algorithms for string-matching and palindrome recognition. doi
  8. (1995). Strongly geodesically automatic groups are hyperbolic. doi
  9. Subgroups of small cancellation groups. doi
  10. (2002). The geometry of relative Cayley graphs for subgroups of hyperbolic groups.
  11. (1995). The isomorphism problem for hyperbolic groups I. doi
  12. (2000). Word-hyperbolic groups have real-time word problem. doi

To submit an update or takedown request for this paper, please submit an Update/Correction/Removal Request.