14:00-14:45 Adrien Boyer (IMJ-PRG): Property RD and Boundary Representations for A2 Buildings

15:00-15:45 Julien Marché (ENS – PSL): Action of endomorphisms of free groups on their SL_2-character varieties

16:15-17:00 Greg Patchell (University of Oxford): Selfless Inclusions of C*-Algebras and Quantum Groups

Abstracts:

Greg Patchell: Selfless Inclusions of C*-Algebras and Quantum Groups

Recently, strong asymptotic freeness, or selflessness, in C-algebras has emerged as a powerful technique to prove important regularity properties including simplicity, unique trace, stable rank 1, and strict comparison. In particular, in Fall 2024, Amrutam, Gao, Kunnwalkam Elayavalli, and I showed that the reduced group C-algebras of all hyperbolic groups with trivial finite radical are selfless, which resolved the open problem of strict comparison for the reduced group C-algebra of the free group on two generators. Since then, our result has been expanded to include a much larger class of groups. Work has also begun on isolating selflessness for C-algebras not arising from groups, including the result of Hayes, Kunnawalkam Elayavalli, and Robert on selflessness of the reduced free product of a large class of C-algebras (see also Flores-Klisse-Ó Cobhthaigh-Pagliero). I will introduce the general notion of a selfless inclusion of C-algebras, with which we will see the selflessness of the reduced unitary compact matrix quantum groups. This work is joint with Ben Hayes, Srivatsav Kunnawalkam Elayavalli, and Leonel Robert.

Adrien Boyer: Property RD and Boundary Representations for A2 Buildings

I will discuss an approach based on boundary representations to prove property RD for discrete groups acting properly and cocompactly on affine buildings of type A2. This result is due to Robertson, Ramagge, and Steger in the late 1990s. I will emphasize geometric arguments involving the Furstenberg boundary that can be used to establish property RD, in particular the notion of “foldings” or “confluences,” as suggested by V. Kaimanovich. Along the way, I will also mention a conjecture of Robertson, Ramagge, and Steger concerning the optimal bound, and propose a possible approach toward resolving it. If time permits, I will also explain what happens in the C2 case. This is joint work with Thang Nguyen.

Julien Marché: Action of endomorphisms of free groups on their SL_2-character varieties.

Let phi : F_n->F_n be an endomorphism and let phi^* denote its action on the character variety X_n=Hom(F_n,SL_2(C))/SL_2(C). Cantat-Gelander-Souto raised the question whether phi^* is an automorphism of the affine variety X_n if and only if phi is an automorphism of F_n. I will describe work in progress which relates directly this question to the action of endomorphisms on the outer space CV_n through a compactification of X_n with special valuations.