Groups and Dynamics Seminar
Organizers:
Rostislav Grigorchuk,
Volodia Nekrashevych,
Zoran Šunić, and
Robin Tucker-Drob.
Arman Darbinyan
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Date Time |
Location | Speaker |
Title – click for abstract |
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01/15 3:00pm |
BLOC 628 |
Volodymyr Nekrashevych Texas A&M University |
Penrose tiling, semigroups, and ample groups.
We will discuss the Penrose tiling (like the one found in the Mitchell Physics Building), its definition, and basic properties. We will introduce a natural groupoid acting on the space of marked tilings and the associated inverse semigroup following J. Kellendonk. We will see that this inverse semigroup has a natural contracting self-similarity and is generated by a finite automaton. The topological full group (ample group) of this semigroup (the group of pattern-equivariant permutations of the tiles) has a nice interpretation as a group of rearrangements of the golden triangle. We will see how this interpretation is related to an Anosov automorphism of a 4-dimensional orbifold, and discuss some properties of the full group. |
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01/22 3:00pm |
BLOC 628 |
Zoran Sunik Hofstra University |
Title: On the conjugator between the Collatz map and the shift map
Certain parity functions were used by Terras in his study of the Collatz map. One can combine these into a single function, call it gamma, understood as an isometry (and hence measure preserving bijection) of the ring Z_2 of dyadic integers. The map gamma conjugates the Collatz map T (extended to Z_2) to the one-sided shift map on Z_2 and, thus, T is ergodic on Z_2. Lagarias related properties of gamma to the Collatz conjecture and other conjectures associated to it. For instance, the Collatz Conjecture is equivalent to the claim that gamma(Z^+) is a subset of 1/3*Z, while the conjecture that all orbits of the Collatz map on Z are eventually periodic is equivalent to the claim that gamma(Z_2 intersected with Q) = Z_2 intersected with Q. Bernstein and Lagarias provided some results on the cycle structure of gamma and, based on these results and empirical observations, they stated several conjectures. We revisit the study of gamma, but think of it as a binary tree automorphism. In fact, we start with a more general approach. Namely, if alpha_0 and alpha_1 are two binary tree automorphisms and T is defined by cases, as \alpha_0(u/2) for even u and alpha_1((u-1)/2) for odd u in Z_2, that is, for all w, T(0w)=alpha_0(w) and T(1w)=alpha_1(w), we define gamma as the unique tree automorphism that conjugates T to the shift map and preserves parity. We prove that, if both alpha_0 and alpha_1 are finite state automorphisms (which they are in case of T), then gamma(Z_2 intersected with Q) contains Z_2 intersected with Q. Moreover, if the minimal self-similar group generated by alpha_0 and alpha_1 is contracting then we have equality, that is, gamma(Z_2 intersected with Q) = Z_2 intersected with Q. In the concrete case when gamma is the conjugator of the Collatz map, by using the tools and the language of tree automorphisms we reprove the results of Bernstein and Lagarias and provide some further insights into gamma. We end with some new empirical data and open questions. In particular, we provi |
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01/29 3:00pm |
BLOC 628 |
Michael Yampolsky University of Toronto, Canada |
Computability and computational complexity questions in dynamics
Much of the modern development of the field of dynamical systems is motivated by numerical experiments. I will discuss whether the modern paradigm of numerical study of dynamics is on a valid theoretical footing. The answers to the question of what can and cannot be computed are mathematically subtle, and can be unexpected even in the simplest case of one-dimensional quadratic maps of the interval. |
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02/05 3:00pm |
BLOC 628 |
Denis Gaidashev Uppsala University, Sweden |
Renormalization and wild attractors for Fibonacci maps
A Fibonacci map is a piecewise defined map of a subset of an interval I onto I with a unique critical point of order d whose orbit undergoes nearest returns at Fibonacci times. It has been shown by Bruin, Keller, Nowicki and van Strien that such maps exhibit wild attractors: Cantor sets of zero Lebesgue measure whose basin of attraction is meager but has positive Lebesgue measure. We will discuss real renormalization, and a trichotomy for Fibonacci maps, similar to the Avila-Lyubich trichotomy for Feigenbaum Julia sets, which, in particular, allows us to show that Fibonacci maps admit wild attractors for d=5.1, and do not for d=3.9 (and, conjecturally, for 2 < d < = 3.9). |
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02/12 3:00pm |
BLOC 628 |
Volodymyr Nekrashevych Texas A&M University |
Penrose tiling, semigroups, and ample groups (part II)
This is a continuation of the previous talk (Jan 15). We will discuss the Penrose tiling (like the one found in the Mitchell Physics Building), its definition, and basic properties. We will introduce a natural groupoid acting on the space of marked tilings and the associated inverse semigroup following J. Kellendonk. We will see that this inverse semigroup has a natural contracting self-similarity and is generated by a finite automaton. The topological full group (ample group) of this semigroup (the group of pattern-equivariant permutations of the tiles) has a nice interpretation as a group of rearrangements of the golden triangle. We will see how this interpretation is related to an Anosov automorphism of a 4-dimensional orbifold, and discuss some properties of the full group. |
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02/19 3:00pm |
BLOC 628 |
Santiago Radi Severo Texas A&M University |
On the congruence subgroup property of IMG(z^2+i)
The Iterated Monodromy group IMG(z^2+i), together with the Basilica group IMG(z^2-1) are in the corner of the theory of self-similar groups developed in the works of Grigorchuk, Bartholdi, Nekrashevych and other researches. It appeared for the first time in their joint article "From Fractal Groups to Fractal Sets" (2002) as an example of a group with the limit space homeomorphic to the Julia set of the polynomial z^2+i and topologically being a dendrite. Later it was discovered that it has an intermediate growth between polynomial and exponential (K-U.Bux and R.Perez), has a finite L-presentation, and that it shares many other properties similar to the Grigorchuk group. Spectral properties of the group were studied by Grigorchuk, Sunik and Savchuk and are not completely known yet. In the study of groups acting on rooted trees an important property is the Congruence Subgroup Property (CSP). If it holds then the closure of the group in the group Aut(T) of automorphisms of the tree coincides with the profinite completion. Absence of CSP makes the study of the profinite completion more difficult. In my talk I will show that IMG(z^2+i) does not satisfy CSP. For doing this I will inspect some important properties of the subgroup lattice of IMG(z^2+i). |
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03/19 3:00pm |
BLOC 628 |
Santiago Radi Severo Texas A&M University |
On finite generation, the congruence subgroup property and just-infiniteness in groups of finite type
Groups of finite type (also known as finitely constrained groups) are closed subgroups of Aut(T), the automorphism group of a regular rooted tree T, whose action locally around every vertex is determined by a finite group of allowed actions. They were introduced in 2005 by Grigorchuk, who proved that the closure of regular branch groups belongs to this class. In 2006, Sunic proved the converse. In the study of groups acting on rooted trees, three important notions play a significant role: the congruence subgroup property (CSP), just-infiniteness (j-oo) and topologically finitely generation (tfg). For instance, if CSP holds, then the group is isomorphic to its profinite completion.
In my talk, I will prove that these three notions are equivalent for groups of finite type that satisfy the so-called Property (E), a property that will be developed in the talk and seems to be true for any group of finite type. As a consequence of this result, it will be shown that the Hanoi tower group in 3 pegs, a group introduced in 2006 by Grigorchuk and Sunic, and known not to be just-infinite, unexpectedly has a just-infinite closure. |
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03/26 3:00pm |
BLOC 628 |
Yuri Bahturin Memorial University of Newfoundland |
On the growth and cogrowth of subgroups in free groups and subalgebras in free Lie algebras
The talk is based on the results of A. Yu. Olshanskii and the speaker. Important results about the growth and cogrowth of subgroups in free groups have been obtained by R. I. Grigorchuk. In this talk I will speak about the special features of the growth and cogrowth when the subgroups are subnormal and the subalgebras are subideals. One says that a subgroup H of a group G is subnormal if there is a descending chain of subgroups G=H_0>H_1>...>H_k=H with every term being a normal subgroup in the previous one. Similarly one defines subideals in an algebra. Thus the growth and cogrowth then give a picture intermediate between these values, say, for normal subgroups and general finitely generated subgroups. |
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04/02 3:00pm |
BLOC 628 |
Ricky Martua Simon Simanjuntak Indiana University Bloomington |
Understanding the Boundary of the Cubic Main Hyperbolic Component
Milnor in 2014 outlined a program to understand the boundary of hyperbolic component, with computational evidence indicating in a family with two free critical points, they are not locally connected. We focus on H_3 the cubic main hyperbolic component. Our approach is to use model space X with a homeomorphism \beta: H_3 -> X, compactify X and study the extension property of \beta to navigate the boundary of H_3. The model space of our choosing is Blaschke product, and the compactification is through measure outlined by McMullen. Our conclusions are the following: (1) Following Petersen-Tan Lei, divide \partial H_3 into “tame” = multiplier of fixed point in disk, and “wild” = multiplier in circle. (2) The tame part is homeomorphic to solid torus (in progress). (3) The wild part is a fractal fibered over Mobius strip, with blow-up at every rational point. (4) The main parabolic component is contained in the wild part. We also present a new result in the theory of Blaschke product: complete understanding the topology of parameter space of Blaschke product. This work is under the supervision of Kevin Pilgrim. |
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04/30 2:00pm |
BLOC 628 |
Alain Valette Université de Neuchâtel |
Reciprocal hyperbolic elements in PSL_2(Z)
An element A in PSL_2(Z) is hyperbolic if |Tr(A)|>2. The maximal virtually abelian subgroup of PSL_2(Z) containing A is either infinite cyclic or infinite dihedral; say that A is reciprocal if the second case happens (A is then conjugate to its inverse). We give a characterization of reciprocal hyperbolic elements in PSL_2(Z) in terms of the continued fractions of their fixed points in P^1(R) (those are quadratic irrationals). Doing so we revisit results of P. Sarnak (2007) and C.-L. Simon (2022), themselves rooted in classical work by Gauss and Fricke & Klein. |
Topics
GENERAL PROBLEMS Burnside
Problem on torsion groups, Milnor Problem on growth, Kaplanski
Problems on zero divisors, Kaplanski-Kadison Conjecture on
Idempotents, and other famous problems of Algebra, Low-Dimensional
Topology, and Analysis, which have algebraic roots.
GROUPS AND GROUP ACTIONS Group actions on trees
and other geometric objects, lattices in Lie groups, fundamental groups of
manifolds, and groups of automorphisms of various structures. The key
is to view everything from different points of view: algebraic,
combinatorial, geometric, and probabalistic.
RANDOMNESS Random walks on groups, statistics on
groups, and statistical models of physics on groups and graphs (such as
the Ising model and Dimer model).
COMBINATORICS Combinatorial properties of
finitely-generated groups and the geometry of their Caley graphs and
Schreier graphs.
GROUP BOUNDARIES Boundaries of
finitely generated groups: Freidental, Poisson, Furstenberg, Gromov,
Martin, etc., boundaries.
AUTOMATA Groups, semigroups, and finite
(and infinite) automata. This includes the theory of formal languages,
groups generated by finite automata, and automatic groups.
DYNAMICS Connections between group theory and
dynamical systems (in particular the link between fractal groups and
holomorphic dynamics, and between branch groups and substitutional
dynamical systems).
FRACTALS Fractal mathematics, related to
self-similar groups and branch groups.
COHOMOLOGY Bounded cohomology, L^2 cohomology, and
their relation to other subjects, in particular operator algebras.
AMENABILITY Asymptotic properties such as
amenability and superamenability, Kazhdan property T, growth, and cogrowth.
ANALYSIS Various topics in Analysis related to
groups (in particular spectral theory of discrete Laplace operators on
graphs and groups).