Dynamical systems and geometric group theory

Question

I am a undergrad student and, at this moment, I am finishing the last semester of my major. Therefore, my search for subjects to study in my Masters has recently started. I have for my whole major being in love with geometry and, in particular, geometric group theory. The problem is that my advisor just retired and my new advisor is heavily focused on dynamical systems, which is an area that I've learned to love.

What I have to ask is: are there any intersections between the two subjects? I really wanted to study GGT, and it would be nice to go to him with a few ideal for a masters project.

Answer 1

I recommend the book Thurston's Work on Surfaces. Heavy on geometry and topology, lot's of interesting dynamical systems stuff, and recommended reading for any student of geometric group theory. (Reading this book launched my own mathematical career, which has lots of mixture of geometric group theory and dynamical systems).

Answer 2

Here is my suggestion. Consider at least browsing the book

"Ergodic theory, symbolic dynamics, and hyperbolic spaces" (Trieste, 1989). Oxford Sci. Publ., Oxford Univ. Press, New York, 1991.

The book is a collection of articles mostly about the connections between the two areas (DS and GGT). It is (quite a bit) dated by now, but, given your level, it might be appropriate. If, after reading the book, you want to see something more modern, take a look here:

Calegari, Danny, The ergodic theory of hyperbolic groups., Hodgson, Craig D. (ed.) et al., Geometry and topology down under. A conference in honour of Hyam Rubinstein, Melbourne, Australia, July 11–22, 2011. Proceedings. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-8480-5/pbk). Contemporary Mathematics 597, 15-52 (2013). ZBL1283.20052.