Olga Kharlampovich

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

Groups acting on trees

Mini course by

Olga Kharlampovich


City University of New York, Hunter College and Graduate Center

May 3, 5, 7, 2021

For the recording of the first lecture, click here, with the password 2VppPyAm

For the recording of the second lecture, click here, with the password dXDEN9NP

For the recording of the third lecture, click here, with the password Ykb3EpXb

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 $\Lambda$ -trees, in particular Zn-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.

  1. Actions on simplicial trees. Amalgamated products and HNN extensions, Bass-Serre theory, graphs of groups. Action of SL2(Z) on the hyperbolic plane.
  2. 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.
  3. Ordered abelian groups $\Lambda$. Actions on $\Lambda$-trees. Structure theorems for finitely generated groups acting freely on Zn-trees and Rn-trees. Finitely presented groups acting freely on $\Lambda$-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, 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, $\Lambda$-trees, World Scientific, 2001.
  • - O.Kharlampovich, A. Miasnikov, D. Serbin, Actions, length functions and non-Archimedian words, IJAC, V. 23, No 2, 2013, 325-455.
  • - O.Kharlampovich, A. Vdovina, Beyond Serre's \Trees" in two directions: $\Lambda$-trees and products of trees, arXiv:1710.10306, 2017.
  • -https://en.wikipedia.org/wiki/Bass-Serre theory (very good article)


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

Meeting address:


or click here.

Webex meeting number (access code): 132 218 0998
Meeting password: t8Zf3wJf8Sm

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