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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].

##### MSC:
 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
##### Keywords:
Bruhat-Tits tree; convolution algebra; homology groups