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Vector bundles and arithmetical groups. I. (English. Russian original) Zbl 0880.20022

Proc. Steklov Inst. Math. 208, 212-233 (1995); translation from Tr. Mat. Inst. Steklova 208, 240-265 (1995).
The preliminary section starts with results on local fields \(K\) of dimension 2. A local field \(K\) of dimension 2 is the quotient field of a (complete) discrete valuation ring \({\mathcal O}_K\) with residue field \(\overline K\) a local field of dimension 1, i.e. \(\overline K\) is the quotient field of a discrete valuation ring with residue field \(k\) (of dimension \(0\)). Let \(G=\text{PGL}(V)\), \(V\) be a vector space of dimension \(2\) over \(K\). The constructions of Weyl group (a non-Coxeter group), Bruhat-decomposition, Bruhat-Tits tree for \(G\) over the 1-dimensional local field \(K/\overline K\) are given. The final and main section is devoted to the construction and study of the Bruhat-Tits tree for \(G\) over local field \(K\) of dimension \(2\). Let \(\Delta=\Delta(G,K/\overline K/k)\) and \(\Delta'=\Delta'(G,K/\overline K)\) be the associated simplicial topological spaces for Bruhat-Tits trees and \(|\Delta|,|\Delta'|\) their geometric realisations.
Theorem. (i) \(|\Delta|\) is a connected contractible topological space of dimension \(1\) with a cell structure (but not a CW-complex). (ii) Every \(x\in|\Delta|\) has a neighbourhood homeomorphic to a bouquet of intervals. (iii) \(G\) acts on \(|\Delta|\) by homeomorphisms. (iv) There is a \(G\)-equivariant surjective simplicial map \(\pi\colon\Delta\to\Delta'\) with \(|\pi|\) a continuous map. (v) For \([L]\in\Delta\) corresponding to a lattice \(L\) in \(V,\pi^{-1}[L]\) is isomorphic to \(\Delta'(\text{PGL}(\overline K)^2,\overline K/k)\). (vi) \(\pi\) induces a bijection between apartments of \(\Delta\) and \(\Delta'\). (vii) On the set \(\Delta_0[2]\) of inner points of \(\Delta\), there is a \(G\) invariant metric \(d\) for which \(\pi\) is a distance decreasing map.
For the entire collection see [Zbl 0863.00012].

MSC:

20E42 Groups with a \(BN\)-pair; buildings
20G25 Linear algebraic groups over local fields and their integers
51E24 Buildings and the geometry of diagrams
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
20E08 Groups acting on trees
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