Olga Kharlampovich

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Atlantic Association for Research in the Mathematical Sciences
CRG "Groups, Rings, Lie and Hopf Algebras"
Atlantic Algebra Centre

Groups acting on trees

Mini course by

City University of New York, Hunter College and Graduate Center

May 3 - 7, 2021

From May 3, 2021 to May 7, 2021, Professor O. Kharlampovich from City University of New York will teach a mini course "Groups acting on trees". Due to the current situation caused by the corona virus disease, the mini course will take place virtually.

Short contents of the mini course

Bass-Serre theory relates group actions on trees with decomposing groups as iterated applications of the operations of amalgamated product and HNN extension, via the notion of the fundamental group of a graph of groups.

One of the generalizations of Bass-Serre theory is the theory of isometric group actions on real trees (R-trees) which are metric spaces generalizing the graph-theoretic notion of a tree. Group actions on R-trees arise naturally in geometric topology, as well as in geometric group theory. Asymptotic cones of groups often have a tree-like structure and give rise to group actions on real trees. The use of R-trees and Λ-trees, in particular Z_n-trees, together with Bass-Serre theory, are key tools in the work on the elementary theory of a free group by Kharlampovich-Miasnikov and Sela.

I will talk about the following topics.

Actions on simplicial trees. Amalgamated products and HNN extensions, Bass-Serre theory, graphs of groups. Action of SL₂(Z) on the hyperbolic plane.
R-trees. Rips' theorem: Let G be a finitely generated group with a free action on an R-tree. Then Gis a free product of surface groups and free abelian groups.
Ordered abelian groups Λ. Actions on Λ-trees. Structure theorems for finitely generated groups acting freely on Z_n-trees and R_n-trees. Finitely presented groups acting freely on Λ-trees.

Possible Reading:

On free products with amalgamation: W. Magnus, A. Karrass and D. Solitar, Combinatorial Group Theory, 3rd edition, Dover 1976, Chapter 4 (exposition is very combinatorial and detailed, a lot of exercises).
On HNN-extensions: R. Lyndon and P. Schupp, Combinatorial group theory, Classics in Math., Springer, Chapter IV (a classical book, exposition is combinatorial and detailed, with various applications).
J.-P. Serre, Trees, Springer, 1980. (classic) (online)
O. Bogopolski, Introduction to Group Theory, EMS, Textbooks in Mathematics, 2008.
M. Bestvina and M. Feigin, Stable actions of groups on real trees, Invent. Math. 121 (1995), 287-321.
M. Bestvina, R-trees in topology, geometry and group theory, 1999. (online).
A different approach: D. Gaboriau, G. Levitt, and F. Paulin, Pseudogroups of isometries of R and Rips Theorem on free actions on R-trees, Israel. J. Math. 87 (1994), 403-428.
I. Chiswell, Introduction to Λ-trees, World Scientific, 2001.
O. Kharlampovich, A. Miasnikov and D. Serbin, Actions, length functions and non-Archimedian words, Internat. J. Algebra Comput. 23 (2013), 325-455.
O.Kharlampovich and A. Vdovina, Beyond Serre's Trees in two directions: Λ-trees and products of trees, arXiv:1710.10306, 2017.
https://en.wikipedia.org/wiki/Bass-Serre theory (very good article)

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Recordings of the mini course:

The course will be suitable for undergraduates, graduate students, postdocs, faculty, and anyone interested in algebra.

The mini course took place via Webex from 10 to 10:50 Eastern Standard Time, which is 11:30 - 12:20 Newfoundland Standard Time.