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Linear and projective boundary of nilpotent groups
Bernhard Krön, Jörg Lehnert, Norbert Seifter and Elmar Teufl
We introduce and study a general concept of boundaries of metric spaces which is based on the principle that sequences tend to the same point at infinity whenever they stay sublinearly close to each other with respect to some reference point. Such a boundary is determined by an arbitrarily chosen family of unbounded subsets.
Examples are the boundary of a CAT(0) space, for which the chosen sets are geodesic rays, and the concept of so-called bundles in infinite graphs due to Bonnington, Richter and Watkins, where the chosen sets are one-way infinite paths.
Our particular interest lies in those boundaries which we get by choosing as families of unbounded subsets the positive or arbitrary powers, respectively, of non-torsion group elements in a finitely generated group (or more general by choosing positive or arbitrary powers of non-compact group elements in compactly generated, locally compact Hausdorff groups). We determine these boundaries for nilpotent groups as a disjoint union of spheres or as disjoint union of projective spaces, respectively. The number of disjoint spheres or projective spaces is equal to the nilpotency class. The dimensions of the spheres or projective spaces are each one less than the torsion-free rank (or compact-free dimension) of the corresponding commutative quotient obtained from the descending central series.
In addition we apply this concept to locally finite graphs with polynomial growth and random walks on nilpotent groups.