This is a report on the construction of $p$-adic $L$-functions attached to ordinary families of holomorphic modular forms on the unitary groups of $n$-dimensional hermitian vector spaces over $CM$ fields. The results have been obtained over a period of nearly 15 years in joint work with Ellen Eischen, Jian-Shu Li, and Chris Skinner. The $p$-adic $L$-functions specialize at classical points to critical values of standard $L$-functions of cohomological automorphic forms on unitary groups, or equivalently of cohomological automorphic forms on $GL(n)$ that satisfy a polarization condition. When $n = 1$ one recovers Katz's construction of $p$-adic $L$-functions of Hecke characters.[-]

We provide two ways to show that the R. Thompson group $F$ has maximal subgroups of infinite index which do not fix any number in the unit interval under the natural action of $F$ on $(0,1)$, thus solving a problem by D. Savchuk. The first way employs Jones' subgroup of the R. Thompson group $F$ and leads to an explicit finitely generated example. The second way employs directed 2-complexes and 2-dimensional analogs of Stallings' core graphs, and gives many implicit examples. We also show that $F$ has a decreasing sequence of finitely generated subgroups $F>H_1>H_2>...$ such that $\cap H_i={1}$ and for every $i$ there exist only finitely many subgroups of $F$ containing $H_i$.[-]

Rivin conjectured that the conjugacy growth series of a hyperbolic group is rational if and only if the group is virtually cyclic. In this talk I will present the proof (joint with Hermiller, Holt and Rees) that the conjugacy growth series of a virtually cyclic group is rational, and then also confirm the other direction of the conjecture, by showing that the conjugacy growth series of a non-elementary hyperbolic group is transcendental (joint with Antolín). The result for non-elementary hyperbolic groups can be used to prove a formal language version of Rivin's conjecture for any finitely generated acylindrically hyperbolic group G, namely that no set of minimal length conjugacy representatives of G can be regular.[-]

A subgroup $H$ of an acylindrically hyperbolic groups $G$ is called geometrically dense if for every non-elementary acylindrical action of $G$ on a hyperbolic space, the limit sets of $G$ and $H$ coincide. We prove that for every ergodic measure preserving action of a countable acylindrically hyperbolic group $G$ on a Borel probability space, either the stabilizer of almost every point is geometrically dense in $G$, or the action is essentially almost free (i.e., the stabilizers are finite). Various corollaries and generalizations of this result will be discussed.[-]

There is a classical result saying that, in a finite group, the probability that two elements commute is never between $5/8$ and 1 (i.e., if it is bigger than $5/8$ then the group is abelian). It seems clear that this fact cannot be translated/adapted to infinite groups, but it is possible to give a notion of degree of commutativity for finitely generated groups (w.r.t. a fixed finite set of generators) as the limit of such probabilities, when counted over successively growing balls in the group. This asymptotic notion is a lot more vague than in the finite setting, but we are still able to prove some results concerning this new concept, the main one being the following: for any finitely generated group of polynomial growth $G$, the commuting degree of $G$ is positive if and only if $G$ is virtually abelian.[-]

Erdös and Rényi introduced a model for studying random graphs of a given "density" and proved that there is a sharp threshold at which lower density random graphs are disconnected and higher density ones are connected. Motivated by ideas in geometric group theory we will explain some new threshold theorems we have discovered for random graphs. We will then explain applications of these results to the geometry of Coxeter groups. Some of this talk will be on joint work with Hagen and Sisto; other parts are joint work with Hagen, Susse, and Falgas-Ravry.[-]