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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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##### References:
 [1] Adams S. , Ballmann W. , Amenable isometry groups of Hadamard spaces , Math. Ann. 312 ( 1998 ) 183 - 195 . MR 1645958 | Zbl 0913.53012 · Zbl 0913.53012 [2] Abels H. , Distal affine transformation groups , J. reine angew. Math. 299 ( 1978 ) 294 - 300 . Article | MR 470944 | Zbl 0367.20047 · Zbl 0367.20047 [3] Abels H. , Distal automorphism groups of Lie groups , J. reine angew. Math. 329 ( 1981 ) 82 - 87 . MR 636446 | Zbl 0463.22006 · Zbl 0463.22006 [4] Allcock D. , Carlson J.A. , Toledo D. , The complex hyperbolic geometry of the moduli space of cubic surfaces , J. Algebraic Geometry 11 ( 2002 ) 659 - 724 . MR 1910264 | Zbl 1080.14532 · Zbl 1080.14532 [5] Ash A. , Mumford D. , Rapoport M. , Tai Y.S. , Smooth Compactification of Locally Symmetric Spaces , Lie Groups: History, Frontiers and Applications , vol. IV , Math. Sci. Press , Brookline, MA , 1975 . MR 457437 | Zbl 0334.14007 · Zbl 0334.14007 [6] Baily W.L. , Borel A. , Compactification of arithmetic quotients of bounded symmetric domains , Ann. of Math. 84 ( 1966 ) 442 - 528 . MR 216035 | Zbl 0154.08602 · Zbl 0154.08602 [7] Behr H. , Higher finiteness properties of S-arithmetic groups in the function field case I , in: Müller T.W. (Ed.), Groups: Topological, Combinatorial and Arithmetic Aspects , London Math. Soc. Lecture Notes Series , vol. 311 , Cambridge University Press , 2004 , pp. 27 - 42 . MR 2073345 | Zbl 1099.20022 · Zbl 1099.20022 [8] Bridson M. , Haefliger A. , Metric Spaces of Non-Positive Curvature , Grund. Math. Wiss. , vol. 319 , Springer , 1999 . MR 1744486 | Zbl 0988.53001 · Zbl 0988.53001 [9] Borel A. , Ji L. , Compactifications of Symmetric and Locally Symmetric Spaces , Mathematics: Theory & Applications , Birkhäuser , 2006 . MR 2189882 | Zbl 1100.22001 · Zbl 1100.22001 [10] Burger M. , Mozes S. , $$\text{CAT}(-1)$$-spaces, divergence groups and their commensurators , J. Amer. Math. Soc. 9 ( 1996 ) 57 - 93 . MR 1325797 | Zbl 0847.22004 · Zbl 0847.22004 [11] Burger M. , Mozes S. , Groups acting on trees: from local to global structure , Publ. Math. IHÉS 92 ( 2000 ) 113 - 150 . Numdam | MR 1839488 | Zbl 1007.22012 · Zbl 1007.22012 [12] Borel A. , Linear Algebraic Groups , Graduate Texts in Math. , vol. 126 , Springer , 1991 . MR 1102012 | Zbl 0726.20030 · Zbl 0726.20030 [13] Bourbaki N. , Intégration VII-VIII , Éléments de Mathématique , Hermann , 1963 . MR 453824 [14] Bourbaki N. , Topologie générale I-IV , Éléments de Mathématique , Hermann , 1971 . MR 358652 | Zbl 0249.54001 · Zbl 0249.54001 [15] Bourbaki N. , Algèbre VIII , Éléments de Mathématique , Hermann , 1973 . MR 417224 [16] Bourbaki N. , Groupes et algèbres de Lie IV-VI , Éléments de Mathématique , Masson , 1981 . MR 647314 | Zbl 0483.22001 · Zbl 0483.22001 [17] Brown K.S. , Buildings , Springer , 1989 . MR 969123 | Zbl 0715.20017 · Zbl 0715.20017 [18] Borel A. , Serre J.-P. , Corners and arithmetic groups , Comm. Math. Helv. 48 ( 1973 ) 463 - 491 . MR 387495 | Zbl 0274.22011 · Zbl 0274.22011 [19] Borel A. , Serre J.-P. , Cohomologie d’immeubles et de groupes S -arithmétiques , Topology 15 ( 1976 ) 211 - 232 . MR 447474 | Zbl 0338.20055 · Zbl 0338.20055 [20] Borel A. , Tits J. , Groupes réductifs , Publ. Math. IHÉS 27 ( 1965 ) 55 - 150 . Numdam | MR 207712 | Zbl 0145.17402 · Zbl 0145.17402 [21] Borel A. , Tits J. , Éléments unipotents et sous-groupes paraboliques de groupes réductifs. I , Invent. Math. 12 ( 1971 ) 95 - 104 . MR 294349 | Zbl 0238.20055 · Zbl 0238.20055 [22] Bruhat F. , Tits J. , Groupes réductifs sur un corps local, I. Données radicielles valuées , Publ. Math. IHÉS 41 ( 1972 ) 5 - 251 . Numdam | MR 327923 | Zbl 0254.14017 · Zbl 0254.14017 [23] Borel A. , Tits J. , Homomorphismes abstraits de groupes algébriques simples , Ann. of Math. 97 ( 1973 ) 499 - 571 . MR 316587 | Zbl 0272.14013 · Zbl 0272.14013 [24] Bruhat F. , Tits J. , Groupes réductifs sur un corps local, II. Schémas en groupes. Existence d’une donnée radicielle valuée , Publ. Math. IHÉS 60 ( 1984 ) 197 - 376 . Numdam | MR 756316 | Zbl 0597.14041 · Zbl 0597.14041 [25] Bruhat F. , Tits J. , Schémas en groupes et immeubles des groupes classiques sur un corps local , Bull. Soc. Math. France 112 ( 1984 ) 259 - 301 . Numdam | MR 788969 | Zbl 0565.14028 · Zbl 0565.14028 [26] Canary R.D. , Epstein D.B.A. , Green P. , Notes on notes of Thurston , in: Epstein D.B.A. (Ed.), Analytical and Geometrical Aspects of Hyperbolic Spaces , London Math. Soc. Lecture Notes Series , vol. 111 , Cambridge University Press , 1987 . MR 903850 | Zbl 0612.57009 · Zbl 0612.57009 [27] Conze J.-P. , Guivarc’h Y. , Remarques sur la distalité dans les espaces vectoriels , C. R. Acad. Sci. Paris 278 ( 1974 ) 1083 - 1086 . MR 339108 | Zbl 0275.54028 · Zbl 0275.54028 [28] de Cornulier Y ., Invariant means on projective spaces, Preprint , 2004. [29] Figá-Talamanca A. , Nebbia C. , Harmonic Analysis and Representation Theory for Groups Acting on Homogeneous Trees , London Math. Soc. Lecture Notes Series , vol. 162 , Cambridge University Press , 1991 . MR 1152801 | Zbl 00050232 · Zbl 1154.22301 [30] Furstenberg H. , The structure of distal flows , Amer. J. Math. 85 ( 1963 ) 477 - 515 . MR 157368 | Zbl 0199.27202 · Zbl 0199.27202 [31] Furstenberg H. , Boundary theory and stochastic processes on homogeneous spaces , in: Moore C.C. (Ed.), Harmonic Analysis on Homogeneous Spaces , Proc. Symp. Pure Math. , vol. XXVI , AMS , Providence, RI , 1972 , pp. 193 - 229 . MR 352328 | Zbl 0289.22011 · Zbl 0289.22011 [32] Ghys É. , de la Harpe P. , Sur les groupes hyperboliques d’après Mikhael Gromov , Progr. Math. , vol. 83 , Birkhäuser , 1990 . MR 1086648 | Zbl 0731.20025 · Zbl 0731.20025 [33] Goldman O. , Iwahori N. , The space of p -adic norms , Acta Math. 109 ( 1963 ) 137 - 177 . MR 144889 | Zbl 0133.29402 · Zbl 0133.29402 [34] Gille P. , Unipotent subgroups of reductive groups in characteristic $$p>0$$ , Duke Math. J. 114 ( 2002 ) 307 - 328 . Article | MR 1920191 | Zbl 1013.20040 · Zbl 1013.20040 [35] Guivarc’h Y. , Compactifications of symmetric spaces and positive eigenfunctions of the Laplacian , in: Taylor J.Ch. (Ed.), Topics in Probability and Lie Groups; Boundary Theory , CRM Proc. Lect. Notes , vol. 28 , AMS , 2001 , pp. 69 - 116 . MR 1832435 | Zbl 01665056 · Zbl 1160.58306 [36] Guivarc’h Y. , Ji L. , Taylor J.Ch. , Compactifications of Symmetric Spaces , Progr. Math. , vol. 156 , Birkhäuser , 1998 . MR 1633171 | Zbl 1053.31006 · Zbl 1053.31006 [37] Landvogt E. , A Compactification of the Bruhat-Tits Building , Lecture Notes in Math. , vol. 1619 , Springer , 1996 . MR 1441308 | Zbl 0935.20034 · Zbl 0935.20034 [38] Lubotzky A. , Mozes S. , Asymptotic properties of unitary representations of tree automorphisms , in: Piccardello M. (Ed.), Harmonic Analysis and Discrete Potential Theory (Frascati, 1991) , Plenum Press , New York , 1992 , pp. 289 - 298 . MR 1222467 [39] Lubotzky A. , Mozes S. , Zimmer R.J. , Superrigidity for the commensurability group of tree lattices , Comm. Math. Helv. 69 ( 1994 ) 523 - 548 . MR 1303226 | Zbl 0839.22011 · Zbl 0839.22011 [40] Mackey G.W. , Induced representations of locally compact groups I , Ann. of Math. 55 ( 1952 ) 101 - 139 . MR 44536 | Zbl 0046.11601 · Zbl 0046.11601 [41] Margulis G.A. , Discrete Subgroups of Semisimple Lie Groups , Ergeb. Math. Grenzgeb. (3) , vol. 17 , Springer , 1991 . MR 1090825 | Zbl 0732.22008 · Zbl 0732.22008 [42] Moore C.C. , Compactifications of symmetric spaces , Amer. J. Math. 86 ( 1964 ) 201 - 218 . MR 161942 | Zbl 0156.03202 · Zbl 0156.03202 [43] Moore C.C. , Amenable subgroups of semisimple groups and proximal flows , Israel J. Math. 34 ( 1979 ) 121 - 138 . MR 571400 | Zbl 0431.22014 · Zbl 0431.22014 [44] Platonov V. , Rapinchuk A. , Algebraic Groups and Number Theory , Pure Appl. Math. , vol. 139 , Academic Press , 1994 . MR 1278263 | Zbl 0841.20046 · Zbl 0841.20046 [45] Prasad G. , Elementary proof of a theorem of Bruhat-Tits-Rousseau and of a theorem of Tits , Bull. Soc. Math. France 110 ( 1982 ) 197 - 202 . Numdam | MR 667750 | Zbl 0492.20029 · Zbl 0492.20029 [46] Raghunathan M.S. , Discrete Subgroups of Lie Groups , Ergeb. Math. Grenzgeb. , vol. 68 , Springer , 1972 . MR 507234 | Zbl 0254.22005 · Zbl 0254.22005 [47] Ronan M.A. , Lectures on Buildings , Perspectives in Mathematics , vol. 7 , Academic Press , 1989 . MR 1005533 | Zbl 0694.51001 · Zbl 0694.51001 [48] Rousseau G ., Immeubles des groupes réductifs sur les corps locaux, thèse d’État , Université de Paris-Sud (Orsay), 1977. MR 491992 | Zbl 0412.22006 · Zbl 0412.22006 [49] Satake I. , On compactifications of the quotient spaces for arithmetically defined discontinuous groups , Ann. of Math. 72 ( 1960 ) 555 - 580 . MR 170356 | Zbl 0146.04701 · Zbl 0146.04701 [50] Satake I. , On representations and compactifications of symmetric Riemannian spaces , Ann. of Math. 71 ( 1960 ) 77 - 110 . MR 118775 | Zbl 0094.34603 · Zbl 0094.34603 [51] Springer T.A. , Linear Algebraic Groups , Progr. Math. , vol. 9 , Birkhäuser , 1998 . MR 1642713 | Zbl 0453.14022 · Zbl 0453.14022 [52] Tits J. , Free subgroups in linear groups , J. Algebra 20 ( 1972 ) 250 - 270 . MR 286898 | Zbl 0236.20032 · Zbl 0236.20032 [53] Tits J. , Reductive groups over local fields , in: Borel A. , Casselman W.A. (Eds.), Automorphic Forms, Representations and L -Functions , Proc. Symp. Pure Math. (Oregon State Univ., Corvallis, 1977), part 1 , vol. XXXIII , AMS , Providence, RI , 1979 , pp. 29 - 69 . MR 546588 | Zbl 0415.20035 · Zbl 0415.20035 [54] Werner A. , Compactification of the Bruhat-Tits building of PGL by lattices of smaller rank , Doc. Math. 6 ( 2001 ) 315 - 341 . MR 1871666 | Zbl 1048.20014 · Zbl 1048.20014 [55] Werner A. , Compactification of the Bruhat-Tits building of PGL by seminorms , Math. Z. 248 ( 2004 ) 511 - 526 . MR 2097372 | Zbl 02131992 · Zbl 1121.20024 [56] Zimmer R.J. , Ergodic Theory and Semisimple Groups , Monographs Math. , vol. 81 , Birkhäuser , 1984 . MR 776417 | Zbl 0571.58015 · Zbl 0571.58015
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