1. Higher rank lattices are not coarse median
- Author
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Thomas Haettel, Institut de Mathématiques et de Modélisation de Montpellier (I3M), Centre National de la Recherche Scientifique (CNRS)-Université Montpellier 2 - Sciences et Techniques (UM2)-Université de Montpellier (UM), Institut de Mathématiques et de Modélisation de Montpellier ( I3M ), and Université Montpellier 2 - Sciences et Techniques ( UM2 ) -Université de Montpellier ( UM ) -Centre National de la Recherche Scientifique ( CNRS )
- Subjects
[ MATH.MATH-GT ] Mathematics [math]/Geometric Topology [math.GT] ,higher rank lattice ,CAT(0) cube complex ,Spherical type ,Rank (differential topology) ,01 natural sciences ,Combinatorics ,51F99 ,Mathematics::Group Theory ,Mathematics - Metric Geometry ,Simple (abstract algebra) ,Lattice (order) ,[MATH.MATH-GT]Mathematics [math]/Geometric Topology [math.GT] ,building ,FOS: Mathematics ,0101 mathematics ,20F65 ,coarse geometry ,CAT (0) cube complex ,Local field ,Mathematics ,Group (mathematics) ,010102 general mathematics ,Metric Geometry (math.MG) ,53C35 ,quasi-isometry ,010101 applied mathematics ,20F65,53C35,51E24,51F99 ,symmetric space ,51E24 ,20F65, 53C35, 51E24, 51F99 ,median algebra ,Geometry and Topology ,Affine transformation ,Cube - Abstract
We show that symmetric spaces and thick affine buildings which are not of spherical type $A_1^r$ have no coarse median in the sense of Bowditch. As a consequence, they are not quasi-isometric to a CAT(0) cube complex, answering a question of Haglund. Another consequence is that any lattice in a simple higher rank group over a local field is not coarse median., 13 pages, 2 figures. To appear in Algebraic & Geometric Topology
- Published
- 2014
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