Spring 2008 RFG Graduate Seminar:
Self-Similar and Branch Groups

The theme for this seminar will be self-similar groups and the related class of branch groups.

Each week a student will present material on a topic related to the semester's theme. The talks will have lax time limits, and frequent interruptions for questions are encouraged.

• branch groups, self-similar groups, iterated monodromy groups, etc.
• amenability of many automata groups
• groups of intermediate growth
• relations to dynamical systems
• Inverse limits of branched coverings over the 2-sphere

Schedule:

First two meetings, on `random' groups (a continuation of the Winter RFG seminar):

1. Friday, 4 April, 3:10-4:00 (Moon)
2. Tuesday, 8 April (Eric)
   
   

   Remaining meetings, on self-similar groups and branch groups (schedule tentative)

3. Definitions: rooted trees, group actions, and automata (Gabe, on Thursday 17 April)
4. Definitions: wreath products, self-similar groups, and branch groups (Pamela, on Tuesday 22 April)
5. Examples: Grigorchuk groups, adding machine, Lamplighter group, Baumslag-Solitar group, etc. (James, on Tuesday, 29 April)
6. Growth and groups of intermediate growth (Joshua, on Tuesday, 6 May)
7. Guest Speaker: Zoran Šunić (Colloquium on Wednesday, 14 May at 3:10, in 1147MSB)
8. Guest Speaker: Zoran Šunić (RFG talk on Thursday, 15 May)
9. Relating dynamics and groups: Sullivan's dictionary (Misha, on Tuesday, 20 May)
10. Self-similar groups and topological dynamics (Jerry, on Tuesday, 27 May)
11. Limit spaces and iterated monodromy groups (Shinpei, on Tuesday, 3 June)

Resources:

• Self-similar groups, a book by Volodymyr Nekrashevych (2005). Preface, Chapter 1. This book is well written and covers self-similar groups from the perspective of dynamical systems. Emphasized are limit spaces and iterated monodromy groups.
  
• Branch Groups, by Laurent Bartholdi, Rostislav Grigorchuk, and Zoran Šunić (2005). This paper is an introduction to branch groups.
  
• Groups of intermediate growth: an introduction for beginners, by Rostislav Grigorchuk and Igor Pak (2006).
  
  
• From fractal groups to fractal sets, by Laurent Bartholdi, Rostislav Grigorchuk, and Volodymyr Nekrashevych (2002). A survey paper on self-similar groups, semigroups, their actions, and their relations to classical objects like Julia sets and boundaries.
  
• Automata, groups, limit spaces, and tilings, by Laurent Bartholdi, Andre Henriques, and Volodymyr Nekrashevych. This paper looks at the connections between the concepts in the title. It contains a nice diagrammatic depiction of these relationships. It ends by presenting examples like the lamplighter group and Baumslag-Solitar groups in the language of `square descriptions' associated to automata.
  
• On amenability of automata groups, by Laurent Bartholdi, Vadim Kaimanovich, and Volodymyr Nekrashevych (2008). As the title suggests, this paper examines amenability of self-similar groups. It proves amenability for some classes of groups generated by finite automata. This paper nicely relates the RFG topics for the Winter and Spring quarters.
  
• Asymptotic aspects of Schreier graphs and Hanoi Towers groups, by Rostislav Grigorchuk and Zoran Šunić (2006). An interesting paper, which introduces groups based on the famous Hanoi Towers Problem.
  
• Self-similarity and branching in group theory, by Rostislav Grigorchuk and Zoran Šunić (2007). A survey of the material in the title, based on a series of talks given by one of the authors.
  
• Iterated monodromy groups, by Volodymyr Nekrashevych (2006). This paper was, I think, the paper where iterated monodromy groups were introduced.
  
• Symbolic dynamics and self-similar groups, by Volodymyr Nekrashevych (2008). This paper discusses groups associated to dynamical systems presented via symbolic presentations. In particular, this paper discusses Julia sets and other curves, including plane-filling curves like the Peano curve and the Sierpinski curve.
  
• Quasiconfromal homeomorphisms and dynamics, I, by Sullivan (1985). This paper was one of the sources for Misha's talk, and examines the dictionary between iterations of functions and Kleinian groups.
• Classification of conformal dynamical systems, by Curtis McMullen (2003). Also referenced by Misha's talk on Sullivan's dictionary. <\ul>