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LINEAR AND PROJECTIVE BOUNDARY OF NILPOTENT GROUPS

Part of: Special aspects of infinite or finite groups Graph theory Spaces with richer structures Lie groups

Published online by Cambridge University Press:  22 December 2014

BERNHARD KRÖN ,
JÖRG LEHNERT ,
NORBERT SEIFTER  and
ELMAR TEUFL
BERNHARD KRÖN
Affiliation:
Fakultät für Mathematik, Universität Wien, Nordbergstraße 15, 1090 Vienna, Austria e-mail: bernhard.kroen@univie.ac.at
JÖRG LEHNERT
Affiliation:
Max Planck Institute for Mathematics in the Sciences, Inselstraße 22, 04103 Leipzig, Germany e-mail: lehnert@mis.mpg.de
NORBERT SEIFTER
Affiliation:
Department Mathematik und Informationstechnologie, Montanuniversität Leoben, Franz-Josef-Straße 18, 8700 Leoben, Austria e-mail: seifter@unileoben.ac.at
ELMAR TEUFL
Affiliation:
Mathematisches Institut, Universität Tübingen, Auf der Morgenstelle 10, 72076 Tübingen, Germany e-mail: elmar.teufl@uni-tuebingen.de
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Abstract

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We define a pseudometric on the set of all unbounded subsets of a metric space. The Kolmogorov quotient of this pseudometric space is a complete metric space. The definition of the pseudometric is guided by the principle that two unbounded subsets have distance 0 whenever they stay sublinearly close. Based on this pseudometric we introduce and study a general concept of boundaries of metric spaces. Such a boundary is the closure of a subset in the Kolmogorov quotient determined by an arbitrarily chosen family of unbounded subsets. Our interest lies in those boundaries which we get by choosing unbounded cyclic sub(semi)groups of a finitely generated group (or more general of a compactly generated, locally compact Hausdorff group). We show that these boundaries are quasi-isometric invariants and determine them in the case of nilpotent groups as a disjoint union of certain spheres (or projective spaces). In addition we apply this concept to vertex-transitive graphs with polynomial growth and to random walks on nilpotent groups.

MSC classification

Primary: 20F65: Geometric group theory
Secondary: 54E35: Metric spaces, metrizability 20F18: Nilpotent groups 22E25: Nilpotent and solvable Lie groups 05C63: Infinite graphs
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 57 , Issue 3 , September 2015 , pp. 591 - 632
Copyright
Copyright © Glasgow Mathematical Journal Trust 2014 

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