In this third lecture, I will talk about random groups and their linear representations. Introduced by Gromov in the 80s, random groups provide a wealth of examples of finitely presented groups. With O. Becker and P. Varju, we have undertaken the study of their character variety, that is the moduli space of their representations with values in  SL(2,C) or a more general semisimple algebraic group. I will show how new families of Zariski-dense rigid groups arise this way and what it tells us about the geometry of word maps. The method showcases a delicate interplay between characteristic zero and characteristic p and relies on GRH.

Skein modules were introduced by Józef H. Przytycki as generalisations of the Jones and HOMFLYPT polynomial link invariants in the 3-sphere to arbitrary 3-manifolds. The Kauffman bracket skein module (KBSM) is the most extensively studied of all. However, computing the KBSM of a 3-manifold is known to be notoriously hard, especially over the ring of Laurent polynomials. With the goal of finding a definite structure of the KBSM over this ring, several conjectures and theorems were stated over the years for KBSMs. We show that some of these conjectures, and even theorems, are not true. In this talk I will briefly discuss a counterexample to Marche’s generalisation of Witten’s conjecture. I will show that a theorem stated by Przytycki in 1999 about the KBSM of the connected sum of two handlebodies does not hold. I will also give the exact structure of the KBSM of the connected sum of two solid tori and show that it is isomorphic to the KBSM of a genus two handlebody modulo some specific handle sliding relations. Moreover, these handle sliding relations can be written in terms of Chebyshev polynomials.