Groups and Dynamics Seminar

Date: September 14, 2016

Time: 3:00PM - 4:00PM

Location: BLOC 220

Speaker: Zoran Sunic, Texas A&M University

Title: Hanoi Towers Group II

Abstract: This will be the second in a series of talks devoted to the Hanoi Towers group H which models the famous XIX century game invented by the French mathematician Lucas (yes, the one from the Lucas sequence).
The group H is a finitely generated, self-similar group acting on a rooted ternary tree in such a way that the Schreier graph of the action on level N models the game played with N disks (the vertices represent the possible configurations and the group generators the possible moves). It can also be viewed as a finitely generated group of isometries of the Cantor set.
The group H has many interesting properties in its own right:
- It is the first first known example of a finitely generated branch group that maps onto the infinite dihedral group.
- It is amenable but not elementary amenable group.
- It is the iterated monodromy group of a post-critically finite rational map on the Riemann sphere.
- Its closure is finitely constrained (in the sense of symbolic dynamics on trees).
- Its Hausdorff dimension is irrational and its limit space is the well known Sierpiński gasket.
- It was the first example of a finitely generated branch group with nontrivial rigid kernel.
- Calculations involving finite dimensional permutational representations of H based on the self-similarity of the group lead to calculation of the spectra of the Sierpiński graphs.
- It has exponential growth.
- It contains a copy of every finite 3-group.
- The elements of the group may be described as finite automata.
- ...
Notable subgroups (Apollonian group, intertwined odometers groups), the higher Hanoi Towers groups (related to versions of the game with more than 3 pegs), and other variations will also be discussed.