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We prove that for every finitely generated hyperbolic group $G$, the action of $G$ on its Gromov boundary induces a hyperfinite equivalence relation.

We prove that the compressed word problem and the compressed simultaneous conjugacy problem are solvable in polynomial time in hyperbolic groups. In such problems, group elements are input as words defined by straight-line programs defined over a finite generating set for the group. We prove also that, for any infinite hyperbolic group G, the compressed knapsack problem in G is NP-complete.

In [5], M. Bonk and B. Kleiner proved a rigidity theorem for expanding quasi-M"obius group actions on Ahlfors n-regular metric spaces with topological dimension n. This led naturally to a rigidity result for quasi-convex geometric actions on CAT(-1)-spaces that can be seen as a metric analog to the ``entropy-rigidity" theorems of U. Hamenst"adt and M. Bourdon. Building on the ideas developed in [5], we establish a rigidity theorem for certain expanding quasi-M"obius group actions on spaces with different metric and topological dimensions. This is motivated by a corresponding entropy-rigidity result in the coarse geometric setting.Our analysis of these ``fractal" metric spaces depends heavily on a combinatorial inequality that relates volume to lengths of curves within the space. We extend such inequalities to a broader metric setting and obtain discrete analogs of some results due to W. Derrick. In the process, we shed light on a related question of Y. Burago and V. Zalgaller about pseudometrics on the n-dimensional unit cube.

We study knot groups that admit a presentation with two generators and one relation. We say that a presentation a, b | r is palindromic if r is a palindrome, that is, if r is a word that reads the same forwards and backwards. We study conditions that allow us to change the given presentation to obtain a palindromic presentation. We prove that if the knot group G admits a faithful discrete SL(2,C)-representation then G admits a palindromic presentation., Se estudian grupos de nudos que admiten una presentación con dos generadores y una relación. Decimos que una presentación a, b | r es palindrómica si r es una palabra palíndromo, es decir, r es una palabra que se lee lo mismo de adelante para atrás que de atrás para adelante. Estudiamos condiciones bajo las cuales es posible cambiar la presentación dada para obtener una presentación palindrómica. Probamos que si el grupo G de un nudo admite una representación fiel en un subgrupo discreto de SL(2,C),entonces G admite una presentación palindrómica.

Postprint (author’s final draft)

Markov partitions provide a 'good' mechanism of symbolic dynamics for uniformly hyperbolic systems, forming the classical foundation for the thermodynamic formalism in this setting, and remaining useful in the modern theory. Usually, however, one takes Bowen's 1970's general construction for granted, or restricts to cases with simpler geometry (as on surfaces) or more algebraic structure. This thesis examines several questions on the algorithmic content of (topological) Markov partitions, starting with the pointwise, entropy-like, topological conjugacy invariant known as orbit complexity. The relation between orbit complexity de nitions of Brudno and Galatolo is examined in general compact spaces, and used in Theorem 2.0.9 to bound the decrease in some of these quantities under semiconjugacy. A corollary, and a pointwise analogue of facts about metric entropy, is that any Markov partition produces symbolic dynamics matching the original orbit complexity at each point. A Lebesgue-typical value for orbit complexity near a hyperbolic attractor is also established (with some use of Brin-Katok local entropy), and is technically distinct from typicality statements discussed by Galatolo, Bonanno and their co-authors. Both our results are proved adapting classical arguments of Bowen for entropy. Chapters 3 and onwards consider the axiomatisation and computable construction of Markov partitions. We propose a framework of 'abstract local product structures' - in principle generalising de nitions of Fried ( nitely presented systems) and Adler - supplemented by shadowing properties for our focus on locally maximal hyperbolic sets. Our approach follows computable analysis via representations (TTE), though even on LMHSs it must be seen as partial; we do not treat e.g. topological transitivity, discontinuity results, positive computational information for local (un)stable manifolds, or partition construction after Fried (1987). Nevertheless, under one characterisation of computabi, Thesis (Ph.D.)--University of Western Australia, 2008

We prove that when a countable group admits a nontrivial Floyd-type boundary, then every nonelementary and metrically proper subgroup contains a noncommutative free subgroup. This generalizes the corresponding well-known results for hyperbolic groups and groups with infinitely many ends. It also shows that no finitely generated amenable group admits a nontrivial boundary of this type. This improves on a theorem by Floyd (Floyd, W. J. (1980). Group completions and limit sets of Kleinian groups. Invent. Math. 57: 205-218) as well as giving an elementary proof of a conjecture stated in that same paper. We also show that if the Floyd boundary of a finitely generated group is nontrivial, then it is a boundary in the sense of Furstenberg and the group acts on it as a convergence group., QC 20100525