PhD course University of Pisa

Geometric group theory and dynamical systems

a PhD course at the Department of Mathematics of the University of Pisa (academic year 2025/26)

Period:

From Thursday, Jan 1, 2026 to Saturday, May 30, 2026

Description:

The purpose of this course is to present some aspects of geometric group theory (group growth, residual finiteness, amenability, embeddability into special classes of groups, and soficity) viewed in connection with the notion of surjunctivity (Gottschalk conjecture) and Garden of Eden type theorems (originally introduced in a symbolic dynamical setting) for dynamical systems. This includes, in particular, a study of subshifts and cellular automata over groups, of topological entropy (for continuous actions of amenable groups on compact Hausdorff spaces), and connections with ring theory (Kaplansky's direct finiteness conjecture). Time permitting, we shall also consider Ruelle-Smale spaces, as well as algebraic dynamical systems (with a quick introduction/review of Pontryagin duality for locally compact abelian groups).

Basic References:

[1] T. Ceccherini-Silberstein and M. Coornaert: Cellular Automata and Groups. 2nd Edition. Springer Monographs in Mathematics, Springer, Cham, 2023.

[2] T. Ceccherini-Silberstein and M. Coornaert: Exercises in Cellular Automata and Groups, with a foreword by Rostislav I. Grigorchuk. Springer Monographs in Mathematics, Springer, Cham, 2023.

[3] M. Denker, Ch. Grillenberger, and K. Sigmund: Ergodic theory on compact spaces. LNM 527, Springer Verlag, 1976.

[4] D. Kerr and H. Li: Ergodic theory. Independence and dichotomies. Springer Monographs in Mathematics, Springer, Cham, 2016.

[5] D. Lind and B. Marcus: An introduction to symbolic dynamics and coding, Second edition. Cambridge Mathematical Library, Cambridge University Press, Cambridge, 2021.

[6] K. Schmidt: Dynamical systems of algebraic origin. vol. 128 of Progress in Mathematics, Birkhäuser Verlag, Basel, 1995.