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Geometric Group Theory and Dynamical Systems
a PhD course at the Department of Mathematics of the University of Pisa (academic year 2025/26)
Period: Tue, Feb 24th - Wed April 1st (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).
(1) The CONFIGURATION SPACE A^G (A a finite alphabet set and G a countable group: PRODISCRETE TOPOLOGY and the SHIFT. A metric compatible with the topology. Endomorphisms of the FULL SHIFT (A^G,G) dynamical systems are the so-called CELLULAR AUTOMATA. Memory sets and associated local defining maps. MINIMAL MEMORY. The CURTIS-HEDLUND-LYNDON theorem. The GARDEN OF EDEN THEOREM over Z: we prove the GOE theorem when the universe is the group Z (integers), namely, that if A is a finite alphabet set and T:A^Z --> A^Z is a cellular automaton, the following conditions are equivalent: (a) T is surjective; ent(T(A^Z)) = log|A|; T is pre-injective. Here, ent(X) is the (topological) entropy of an invariant subset X \subset A^Z. (NOTES)
(2) FREE groups: we define free groups, prove their existence, and discuss the normal form (i.e., the REDUCED form) for their elements. (NOTES)
(3) RESIDUALLY FINITE groups: we define and study residual finiteness for groups. Several characterizations are given (including triviality of their RESIDUAL SUBGROUP). We prove F.Klein's PING-PONG lemma and use it to show that free groups are residually finite. We prove W.Lawton's proof of Gottschalk's SURJUNCTIVITY conjecture for residually finite grops, namely, that if A is a finite alphabet set, G is a residually finite group, and T:A^G--> A^G is a cellular automaton, then "T injective" implies "T surjective". (NOTES)
(4) AMENABLE groups: we define and study amenability of groups in terms of (1) existence of INVARIANT FINITELY-ADDITIVE PROBABILITY MEASURES on G (this is von Neumann's original definition); (2) existence of INVARIANT MEANS on G; (3) various forms of the FOLNER CONDITIONS; (4) nonexistence of PARADOXICAL DECOMPOSITIONS. We observe that the free group is not amenable. Several stability properties (e.g., passing to SUBGROUPS/QUOTIENTS, EXTENSIONS of amenable groups by amenable groups) for the class of amenable groups are presented. We discuss BIPARTITE GRAPHS (matchings, Hall's marriage theorem, and its HAREM version) and use these to prove the Folner-Tarski Theorem. Statememt of the ORNSTEIN-WEISS theorem (a sort of generalization of Fekete's subadditivity lemma for amenable groups). (NOTES)
(5) TOPOLOGICAL ENTROPY: we define topological entropy (resp. metric entropy) for actions of a amenable groups on compact (metrizable) spaces - after Adler-Konheim-McAndrew (resp. Bowen and Dinaburg). We recover the definition of ENTROPY ent(X) for X \subset A^G a subshift. (NOTES)
(6) The GARDEN OF EDEN THEOREM for AMENABLE GROUPS: Tilings in groups (and their existence; examples). We outline the proof of the GOE theorem when the universe is an amenable group; we show that if G = F_2, the free group of rank two, both the MOORE property (every surjective cellular automaton T: A^G --> A^G is pre-injective) and the MYHILL property (every pre-injective cellular automaton T: A^G --> A^G is surjective) may fail to hold. We discuss the von Neumann problem (on the existence of nonamenable groups containing no free subgroups) and its solution by A.Yu.Olshansky. The TARSKI NUMBER of a paradoxical group (estimates for the FREE BURNSIDE GROUPS B(m,n) with m \geq 2 and n \geq 665 odd). INDUCTION and RESTRICTION of cellular automata. Bartholdi's theorems (and a new characterization of amenability via SYMBOLIC DYNAMICS). (NOTES)
(7) LOCAL EMBEDDABILITY AND SOFIC GROUPS: we define the notions of local embeddability of a group into a class of groups (typically, the class of FINTE groups and of AMENABLE groups, yielding LEF and LEA groups) Then we introduce and study the notion of soficity for groups. We remind the notions of ULTRAFILTERS and LIMITS along FILTERS and ULTRAPRODUCTS and give a characterization of soficity in terms of nonstandard analysis. (NOTES)
(8) THE WEISS CONDITION (for sofic groups) and THE GROMOV-WEISS SURJUNCTIVITY THEOREM (1st proof): we discuss the Weiss condition for finitely generated groups. A finitely generated group is sofic if and only if it satisfies the Weiss condition. A group is sofic (resp. surjunctive) if and only if it is locally sofic (resp. locally surjunctive). We present a proof of the Gromov-Weiss surjunctivity theorem for finitely generated sofic groups (using the Weiss condition). (NOTES)
(9) SOFIC TOPOLOGICAL DIMENSIONENTROPY and THE GROMOV-WEISS SURJUNCTIVITY THEOREM (2nd proof): we introduce the notion of sofic topological entropy h_\Sigma(X,G) of a dynamical system (X,G) where X is (as usual) a compact metrizable space equipped with the action of a sofic group G (given with a sofic approximation \Sigma). We use pseudometrics and the original approach by Kerr-Li. We discuss the case of subshifts X \subset A^G and prove that h_\Sigma(X,G) \leq \log|A| with equality iff X = A^G. We derive a second proof of the Gromov-Weiss surjunctivity theorem for finitely sofic groups. (NOTES)
(10) KAPLANSKY'S STABLE FINITENESS CONJECTURE and SOFIC MONOIDS: we discuss LINEAR CELLULAR AUTOMATA and define K-surjunctivuty of groups. The algebra LCA(G,V) of linear cellular automata with coefficients in a finite dimensional vector space over a field K is isomorphic to the algebra M_d(K[G]) of dxd matrices (d = dim_K(V)) with coefficinets in the group ring K[G]. We deduce that K[G] is stably finite if and only if G is K-surjunctive. We discuss some basics of FIRST ORDER MODEL THEORY (including the 1st and 2nd PRINCIPLE of LEFSCHETZ) and use it to prove that K[G] is stably finite for G any SURJUNCTIVE GROUP. This (combined with the Gromov-Weiss surjunctvivty theorem) yields another proof of the Elek-Szabo theorem (sofic groups satisfy the Kaplansky conjecture). We also introdice and discuss SOFIC MONOIDS: we provide several examples (including the soficity of the monoid M(S) associated with a semigroup (possibly a monoid itself)). We show that the BICYCLIC MONOID B = <p,q:pq=1> is NOT sofic. (NOTES)
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.
Scheduled lessons:
Tuesday, Feb 24, 2026, 14:00 (120 minutes), Aula Riunioni (Department of Mathematics)
Wednesday, Feb 25, 2026, 14:00 (120 minutes), Aula Riunioni (Department of Mathematics)
Thursday, Feb 26, 2026, 11:00 (120 minutes), Aula Riunioni (Department of Mathematics)
Friday, Feb 27, 2026, 11:00 (120 minutes), Saletta Riunioni
Monday, Mar 2, 2026, 14:00 (120 minutes), Aula Riunioni (Department of Mathematics)
Tuesday, Mar 3, 2026, 11:00 (120 minutes), Aula Riunioni (Department of Mathematics)
Monday, Mar 9, 2026, 14:00 (120 minutes), Aula Riunioni (Department of Mathematics)
Tuesday, Mar 10, 2026, 11:00 (120 minutes), Aula Riunioni (Department of Mathematics)
Monday, Mar 16, 2026, 14:00 (120 minutes), Aula Riunioni (Department of Mathematics)
Tuesday, Mar 17, 2026, 11:00 (120 minutes), Aula Riunioni (Department of Mathematics)
Monday, Mar 23, 2026, 14:00 (120 minutes), Aula Riunioni (Department of Mathematics)
Tuesday, Mar 24, 2026, 11:00 (120 minutes), Aula Riunioni (Department of Mathematics)
Monday, Mar 30, 2026, 14:00 (120 minutes), Aula Riunioni (Department of Mathematics)
Tuesday, Mar 31, 2026, 11:00 (120 minutes), Aula Riunioni (Department of Mathematics)
Wednesday, April 1 2026, 11:00 (120 minutes), Saletta Riunioni
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