arXiv subjects
Publications and source records attributed to Lander Guerrero Sánchez.
We study the asymptotic behaviour of the cohomology of subgroups $Γ$ of an algebraic group $G$ with coefficients in the various irreducible rational representations of $G$ and raise a conjecture about it. Namely, we expect that the dimensions of these cohomology groups approximate the L2-analogues of Betti numbers of $Γ$ with a controlled error term. We provide positive answers when $G$ is a product of copies of $SL_2$.
Let $p\geq 5$ be a prime number, let $n\geq 2$ be a natural number and let $\text{Heis}(p^n)$ denote the Heisenberg group modulo $p^n$. We study the Lyndon-Hochschild-Serre spectral sequence $E(\text{Heis}(p^n))$ associated to $\text{Heis}(p^n)$ considered as a split extension, and show that, $E(\text{Heis}(p^n))$ collapses in the third page. Moreover, for a fixed $p$, the spectral sequences $E(\text{Heis}(p^n))$ are isomorphic from the second page on.
Let p>3 be a prime number and let r be an integer with 1<r<p-1. For each r, let moreover G_r denote the unique quotient of the maximal class pro-p group of size p^{r+1}. We show that the mod-p cohomology ring of G_r has depth one and that, in turn, it satisfies the equalities in Carlson's depth conjecture [3].
This thesis aims to serve as an introduction to the theory of quasitilings for amenable groups. In order to showcase the power of this theory, we focus on the study of the Sofic Lück Approximation Conjecture, which can be proven for amenable groups by making use of quasitilings. The first four chapters of the thesis are an exposition of the aforementioned topics, collected from the literature. After that, in the fifth and final chapter we present some new results related to the Sofic Lück Approximation Conjecture for amenable groups over discrete valuation rings and number fields.