# Bass-Serre theory textbook

I am a PhD freshman working on topological graph theory and geometric group theory. I would like to learn some Bass-Serre theory. What do you think is the best introductory textbook in this topic? Thank you in advance for your recommendations.

• Serre’s book and Scott-Wall’s survey remain by far the best sources. Jun 17, 2020 at 0:30
• I like the beginning of Dunwoody and Dicks book. They might be a little too brief for a first introduction but they have a number of nice applications and I love they prove the universal covering tree is a tree using homology rather than normal forms. Jun 17, 2020 at 16:44

As mentioned by Andy Putman in the comments, the classical (and probably the best) references are Serre's book Trees and Scott and Wall's paper Topological methods in group theory.

Serre's approach is elementary and essentially self-contained, based on combinatorial arguments. Scott and Wall's approach is based on algebraic topology (in particular, they use covering theory). If you are already familiar with some algebraic topology, then you may find Scott and Wall's approach more intuitive, more visual. Personally, I began by reading Serre's book, but I prefer the point of view of graphs of spaces.

You can also take a look at Henry Wilton's blog Geometric group theory. There are several blogposts dedicated to the Bass-Serre theory. In complement to Serre's book, an interesting complement is Higgins' article The fundamental groupoid of a graph of groups. Looking at the fundamental groupoid instead of the fundamental group makes some arguments more natural.

Depending on how far you want to go, there is a very short and very well written note by Wenyuan Yang here : http://bicmr.pku.edu.cn/~wyang/ggt/BassSerre.pdf

You can find the construction of the Bass-Serre tree for Amalgamated sums and HNN extensions and some related theorems.

"Trees" by JP Serre.

• As it's been quoted several times, let me mention that his book "Arbres amalgames $\mathrm{SL}_2$", in French, was published in 1977 by Astérisque, and later (1980) translated into English and published as "Trees" at Springer.
– YCor
Jun 17, 2020 at 8:12