Cayley graphs with few automorphisms: beyond finitely generated groups (Paul-Henry Leemann, Xi'an Jiaotong-Liverpool University)

24.02.2026 10:30

Given a group G, the choice of a generating set S turns G onto a metric space (G,d_S), on which G faithfully acts by left multiplication. While the group of isometries of (G, d_S) is often much larger than G, it can sometimes coincide with G. In such a case, we say that (G,d_S) is called a graphical rigid representation (GRR) of G. The name comes from the fact that Iso(G,d_S) is isomorphic to the automorphism group of the Cayley graph Cay(G,S).
A group G admits a GRR if and only if it can be realised as the automorphism group of a "nice" graph. The classification of groups admitting GRR began in the late 1960s and was completed in 1981 for finite groups. In 2022, alongside Mikael de la Salle, we solved the infinite finitely generated case. In this talk, I will explain how to handle countable (not necessarily finitely generated) groups. While some methods for finitely generated groups remain applicable in this broader context, new ones are also necessary. Specifically, I'll define a "small generating set" for groups of arbitrary cardinality and prove that every group has such a generating set.
The talk will be accessible to a wide audience and no prior knowledge of the subject is required.

Lieu

Bâtiment: Conseil Général 7-9

Room 1-05, Tuesday 24.02.2026, Séminaire "Groupes et géométrie"

entrée libre