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Equivariant homology for \(SL(2)\) of a \(p\)-adic field. (English) Zbl 0844.46043
Fox, Jeffrey (ed.) et al., Index theory and operator algebras. Proceedings of a CBMS regional conference on \(K\)-homology and index theory, held August 6-10, 1991 in Boulder, CO, USA. Providence, RI: American Mathematical Society. Contemp. Math. 148, 1-18 (1993).
Let \(G\) be a locally compact, second countable group that acts simplicially on some tree. Then the \(K\)-groups of the reduced \(C^*\)-algebra \(C^*_r (G)\) can be expressed in terms of those of the same algebras of the vertex and edge stabilizers, via a six-term cyclic exact sequence [M. V. Pimsner, Invent. Math. 86, 603-634 (1986; Zbl 0638.46049)]. The present paper treats the case of \(G= \text{SL}_2 (F)\), the group of unimodular \(2\times 2\) matrices with entries in a \(p\)-adic field \(F\) (i.e., a finite extension of \(Q_p\)), acting on its Bruhat-Tits tree in the sense of J.-P. Serre [Trees, English translation, Berlin (1980; Zbl 0548.20018); French original: ‘Arbres, amalgames, \(\text{SL}_2\)’, Astérisque, Paris 46 (1977; Zbl 0369.20013)]. The homology groups are now very closely related to the cyclic ones of the convolution algebra \(C_c^\infty (G)\) of smooth compactly supported functions on \(G\), and the outcomes of computations are similar to certain purely algebraic considerations of P. Blanc and J.-L. Brylinski [J. Funct. Anal. 109, No. 2, 289-330 (1992; Zbl 0783.55004); also in Lect. Notes Math. 1271, 33-72 (1987; Zbl 0643.16012)].
For the entire collection see [Zbl 0778.00022].

46M20 Methods of algebraic topology in functional analysis (cohomology, sheaf and bundle theory, etc.)
46L80 \(K\)-theory and operator algebras (including cyclic theory)
22E50 Representations of Lie and linear algebraic groups over local fields
22E35 Analysis on \(p\)-adic Lie groups