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arXiv · 2608.25633

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

Isadora Vanzella Picinini
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Abstract

Let $\mathbb{Z}[1/n] := \{a/n^r: a \in \mathbb{Z},r \in \mathbb{Z} \geq 0\}$ 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 $n$.

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Isadora Vanzella Picinini. 2026-08-26. The integral homology of $\text{SL}_2(\mathbb{Z}[1/n])$. https://arxiv.org/abs/2608.25633

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