The homology of \(\text{SL}_ 2(F[t,t^{-1}])\)
Let \(F\) be a number field. The author shows that \(H_k (SL_2 (F[t,t^{-1}]); \mathbb{Q}) \cong H_{k-1} (F^*; \mathbb{Q})\) provided that \(k \geq 2r + 3s + 2\), where \(r\) is the number of real embeddings, \(s\) the number of pairs of complex embeddings. The problem is broken up into manageable pieces by means of the action of \(SL_2(F[t, t^{-1}])\) on a tree and the associated spectral sequence. One of the intermediate results is that for an infinite field of characteristic zero \(H_k (SL_2 (F[t]); \mathbb{Q}) \cong H_k (SL_2(F); \mathbb{Q})\).
scientific article; zbMATH DE number 869801