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Group-theoretic compactification of Bruhat-Tits buildings. (English) Zbl 1126.20029
The authors consider the rational points \(G_F\) of a semisimple group \(G\) over a non-Archimedean local field \(F\) and its Bruhat-Tits building \(X\). Several compactifications of \(X\) are known. Here the authors are concerned with compactifications by group-theoretic techniques which allow them to obtain group-theoretic results like geometric parametrizations of certain classes of subgroups.
The first important step in this direction is a convergence theorem for parabolic subgroups of \(G\) in the Chabauty topology. The map that assigns to a vertex \(x\in X\) its stabilizer in \(G_F\) establishes a \(G_F\)-equivariant topological embedding of the set \(V_X\) of all vertices in \(X\) into the compact space \({\mathcal S}(G_F)\) of closed subgroups of \(G_F\) endowed with the Chabauty uniform structure. The range of this map consists of the set \({\mathcal K}(G_F)\) of all maximal compact subgroups of \(G_F\) and the authors define the closure of \({\mathcal K}(G_F)\) in \({\mathcal S}(G_F)\) as the group-theoretic compactification \(\overline V_X^{\text{gp}}\) of \(X\).
The authors obtain descriptions of Euclidean buildings in the boundary of \(\overline V_X^{\text{gp}}\). These results are used to compare the group-theoretic compactification of \(X\) with the polyhedral compactification \(\overline X^{\text{pol}}\) of \(X\) introduced by E. Landvogt [A compactification of the Bruhat-Tits building. Lect. Notes Math. 1619 (1995; Zbl 0935.20034)]. If \(\overline V_X^{\text{pol}}\) is the closure of \(V_X\) in the polyhedral space \(\overline X^{\text{pol}}\), it is shown that \(\overline V_X^{\text{pol}}\) and \(\overline V_X^{\text{gp}}\) are naturally \(G_F\)-equivariantly homeomorphic; in particular, this establishes that \(\overline V_X^{\text{pol}}\) is a compact \(G_F\)-space.
In a similar way the group-theoretic compactification of a locally finite tree \(X\) (not necessarily coming from a rank-one algebraic group) is defined by using the Chabauty topology on the closed subgroups of a sufficiently transitive automorphism group \(G\). This not only provides examples for the previous constructions and results but also serves as the induction basis when dealing with parametrizations of certain subgroups in the following section. It is shown that the geometric, polyhedral, group-theoretic and measure-theoretic compactifications of the vertex set of a semi-homogeneous tree \(X\) are \(G\)-homeomorphic where \(G\) is a closed locally \(\infty\)-transitive subgroup of \(\operatorname{Aut}(X)\).
Returning to the algebraic situation the authors consider a simply connected semisimple \(F\)-group \(G\) of arbitrary positive \(F\)-rank, where \(F\) is a locally compact non-Archimedean local field, and two classes of subgroups of \(G_F\) for both of which parametrization theorems for the subgroups involved are obtained. In the first class, closed amenable Zariski connected subgroups of \(G_F\) are investigated. Any such subgroup fixes a facet of the polyhedral compactification \(\overline X^{\text{pol}}\) of \(X\) and those subgroups maximal with respect to these properties are the vertex fixators for the \(G_F\)-action on \(\overline X^{\text{pol}}\). This then implies that any closed amenable subgroup of \(G_F\) has a finite orbit in \(\overline X^{\text{pol}}\). In the second class subgroups with distal adjoint action on the Lie algebra of \(G_F\) are considered. Any such group is contained in a point of \(\overline V_X^{\text{gp}}\) and the maximal distal subgroups of \(G_F\) are the groups of \(\overline V_X^{\text{gp}}\), that is, they are maximal compact subgroups and the limit of maximal compact subgroups; in particular, they are closed and Zariski connected.
In the last section the authors use the special linear groups to illustrate the many ideas and results developed in the paper. They also give a proof for the continuity of the action of \(G_F\) on the polyhedral compactification of \(X\) and remedy a gap in Landvogt’s paper [loc. cit.].

MSC:
20G25 Linear algebraic groups over local fields and their integers
20E42 Groups with a \(BN\)-pair; buildings
51E24 Buildings and the geometry of diagrams
20F65 Geometric group theory
22F50 Groups as automorphisms of other structures
22E40 Discrete subgroups of Lie groups
20E08 Groups acting on trees
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