Zoran Sunic

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

Cayley graph automatic groups are not necessarily cayley graph biautomatic

We show that there are Cayley graph automatic groups that are not Cayley graph biautomatic. In addition, we show that there are Cayley graph automatic groups with undecidable Conjugacy Problem and that the Isomorphism Problem is undecidable in the class of Cayley graph automatic groups.

From Self-Similar Groups to Self-Similar Sets and Spectra

The survey presents developments in the theory of self-similar groups leading to applications to the study of fractal sets and graphs and their associated spectra.

Cellular automata between sofic tree shifts

We study the sofic tree shifts of A^@S^^^@?, where @S^@? is a regular rooted tree of finite rank. In particular, we give their characterization in terms of unrestricted Rabin automata. We show that if X@?A^@S^^^@? is a sofic tree shift, then the configurations in X whose orbit under the shift action is finite are dense in X, and, as a consequence of this, we deduce that every injective cellular automata @t:X->X is surjective. Moreover, a characterization of sofic tree shifts in terms of general Rabin automata is given. We present an algorithm for establishing whether two unrestricted Rabin automata accept the same sofic tree shift or not. This allows us to prove the decidability of the surjectivity problem for cellular automata between sofic tree shifts. We also prove the decidability of the injectivity problem for cellular automata defined on a tree shift of finite type.

Normal art galleries: Wall in - all in

We introduce the notion of a normal gallery, a gallery in which any configuration of guards that visually covers the walls necessarily covers the entire gallery. We show that any star gallery is normal and any gallery with at most two reflex corners is normal. A polynomial time algorithm is provided deciding if, for a given gallery and a finite set of positions within the gallery, there exists a configuration of guards in some of these positions that visually covers the walls, but not the entire gallery.

Twin Towers of Hanoi

In the Twin Towers of Hanoi version of the well known Towers of Hanoi Problem there are two coupled sets of pegs. In each move, one chooses a pair of pegs in one of the sets and performs the only possible legal transfer of a disk between the chosen pegs (the smallest disk from one of the pegs is moved to the other peg), but also, simultaneously, between the corresponding pair of pegs in the coupled set (thus the same sequence of moves is always used in both sets). We provide upper and lower bounds on the length of the optimal solutions to problems of the following type. Given an initial and a final position of n disks in each of the coupled sets, what is the smallest number of moves needed to simultaneously obtain the final position from the initial one in each set? Our analysis is based on the use of a group, called Hanoi Towers group, of rooted ternary tree automorphisms, which models the original problem in such a way that the configurations on n disks are the vertices at level n of the tree and the action of the generators of the group represents the three possible moves between the three pegs. The twin version of the problem is analyzed by considering the action of Hanoi Towers group on pairs of vertices.

FINITELY CONSTRAINED GROUPS OF MAXIMAL HAUSDORFF DIMENSION

We prove that if G_P is a finitely constrained group of binary rooted tree automorphisms (a group binary tree subshift of finite type) defined by an essential pattern group P of pattern size d, d>1, and if G_P has maximal Hausdorff dimension (equal to 1-1/2^{d-1}), then G_P is not topologically finitely generated. We describe precisely all essential pattern groups P that yield finitely constrained groups with maximal Haudorff dimension. For a given size d, d>1, there are exactly 2^{d-1} such pattern groups and they are all maximal in the group of automorphisms of the finite rooted regular tree of depth d.