Alternatively have a look at the program.

## Twisted Patterson-Sullivan measure and applications to growth problems

Given a group G acting properly by isometries on a metric space X, the exponential growth rate of G with respect to X measures "how big" the orbits of G are. If H is a subgroup of G, its exponential growth rate is bounded above by the one of G. We are interested in the following question: when do H and G have the same exponential growth rate ?

## Sofic approximations - what’s the problem

I am planning to give a general introduction to sofic groups, mention a few applications to fundamental conjectures about groups and group rings, and explain Misha Gromov’s conjecture that all groups are sofic. Finally I want to present a natural generalization of Gromov’s conjecture due to Laszlo Lovasz and Balasz Szegedy that has recently been disproved in joint work with Gabor Kun.

## Cross ratios on cube complexes and length-spectrum rigidity

Cross ratios naturally arise on boundaries of negatively curved spaces

and are a valuable tool in their study. If one however slightly relaxes the

curvature assumption, simply requiring it to be *non-positive*, things

tend to get more complicated. Even the mere definition of a cross ratio becomes a

more delicate matter.

Restricting to the context of CAT(0) cube complexes $X$, we observe that

most issues disappear if one considers the $\ell^1$ metric on $X$, rather than the

CAT(0) metric. We obtain a canonical cross ratio on the horoboundary of the $\ell^1$

## RAAGs and Stable Commutator Length

Stable commutator length ($\mathrm{scl}$) is a well established invariant of elements $g$ in the

commutator subgroup (write $\mathrm{scl}(g)$) and has both geometric and algebraic meaning.

## Property (T) for $\mathrm{Aut}(F_n)$

I will present a recent proof, by Kaluba, Nowak, and myself, of the fact that $\mathrm{Aut}(F_n)$ has Kazhdan's property (T) for every $n>4$. I will discuss how the same strategy gives a new proof of property (T) for $\mathrm{SL}_n(\mathbb Z)$.

## Fixed point properties for group actions on Banach spaces

Group actions on Banach spaces (in particular on Lp-spaces) have seen a growing interest in the last decades. After an introduction to this topic, I will explain a joint work with Mikael de la Salle, in which we established a spectral criterion for fixed point properties of group actions on large classes of Banach spaces (including Lp-spaces). This criterion can be applied to random groups in certain models. Our work also lead to new estimates on the conformal dimension of the boundary of random groups.

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