Groups St Andrews 2005: Self-similarity and branching in group theory

Let G = SL(2, Qp). Let k > 2 and consider the space Hom(Fk, G) where Fk is the free group on k generators. This space can be thought of as the space of all marked k-generated subgroups of G, i.e., subgroups with a given set of k generators. There is a natural action of the group Aut(Fk) on Hom(Fk, G) by pre-composition. I will prove that this action is ergodic on the subset of dense subgroups. This means that every measurable property either holds or fails to hold for almost all k-generated subgroups of G together. Speaker: Volodymyr Nekrashevych (Texas A&M) Title: Self-similar groups, limit spaces and tilings Abstract: We explore the connections between automata, groups, limit spaces of self-similar actions, and tilings. In particular, we show how a group acting “nicely” on a tree gives rise to a self-covering of a topological groupoid, and how the group can be reconstructed from the groupoid and its covering. The connection is via finite-state automata. These define decomposition rules, or self-similar tilings, on leaves of the solenoid associated with the covering. Speaker: Olga Kharlampovich (McGill) Title: Undecidability of Markov Properties Abstract: A group-theoretic property P is said to be a Markov property if it is preserved under isomorphism and if it satisfies: 1. There is a finitely presented group which has property P . 2. There is a finitely presented group which cannot be embedded in any finitely presented group with property P . Adyan and Rabin showed that any Markov property cannot be decided from a finite presentation. We give a survey of how this is proved. Speaker: Alexei Miasnikov (McGill) Title: The conjugacy problem for the Grigorchuk group has polynomial time complexity Abstract: We discuss algorithmic complexity of the conjugacy problem in the original Grigorchuk group. Recently this group was proposed as a possible platform for cryptographic schemes (see [4, 15, 14]), where the algorithmic security of the schemes is based on the computational hardness of certain variations of the word and conjugacy problems. We show that the conjugacy problem in the Grigorchuk group can be solved in polynomial time. To prove it we replace the standard length by a new, weighted length, called the norm, and show that the standard splitting of elements from St(1) has very nice metric properties relative to the norm. Speaker: Mark Sapir (Vanderbilt) Title: Residual finiteness of 1-related groups Abstract: We prove that with probability tending to 1, a 1-relator group with at least 3 generators and the relator of length n is residually finite, virtually residually (finite p)-group for all sufficiently large p, and coherent. The proof uses both combinatorial group theory, non-trivial results about Brownian motions, and non-trivial algebraic geometry (and Galois theory). This is a joint work with A. Borisov and I. Kozakova. Speaker: Dmytro Savchuk (Texas A&M) Title: GAP package AutomGrp for computations in self-similar groups and semigroups: functionality, examples and applications Abstract: Self-similar groups and semigroups are very interesting from the computational point of view because computations related to these groups are often cumbersome to be performed by hand. Many algorithms related to these groups were implemented in AutomGrp package developed by the authors (available at http://www.gap-system.org/Packages/automgrp.html). We describe the functionality of the package, give some examples and provide several applications. This is joint with Yevgen Muntyan Speaker: Benjamin Steinberg (Carleton) Title: The Ribes-Zalesskii Product Theorem and rational subsets of groups