The integral homology of ${\text{SL}}_2(\mathbb{Z}[1/n])$

Abstract

Let $\mathbb{Z}[1/n]:=\{a/n^r:a\in \mathbb{Z},r\in \mathbb{Z}\ge 0\}$ be the euclidean domain obtained from the ring of integers $\mathbb{Z}$ by localizing at $n$. Moreover, let ${\text{SL}}_2(\mathbb{Z}[1/n])$ be the group of all invertible 2-by-2 matrices of determinant one with entries in $\mathbb{Z}[1/n]$. The homology groups $H_k({\text{SL}}_2(\mathbb{Z}[1/n]),\mathbb{Z})$ are of interest in Geometric Group Theory, Algebraic Number Theory and Algebraic K-theory. In this dissertation, we study these groups via a spectral sequence derived by the action of ${\text{SL}}_2(\mathbb{Z}[1/n])$ on the product of the trees $B_p$ associated with the $p$-adic valuation on $\mathbb{Q}$ where $p$ is a prime factor of $n$.

128 pages, 13 figures