Linear and Projective Boundary of Nilpotent Groups

Abstract

Abstract. 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.

Linear and projective boundaries in HNN-extensions and distortion phenomena

“…A new concept of quasi-isometry invariant boundaries of metric spaces has recently been introduced by Krön, Lehnert, Seifter and Teufl [6]. It is related to a concept due to Bonnington, Richter and Watkins [1].…”

“…(see [2,Section 9]) fits into it, after a small modification. See [6] for a more detailed discussion of this relationship.…”

“…Moreover, by definition it is clear that the diameter of LG and of P G is at most 1. For more details we refer to [6].…”

“…The linear boundary of finitely generated nilpotent groups is (homeomorphic to) the disjoint union of spheres with dimensions d i , which correspond to the free abelian quotients of rank d i C 1 in the central series, and the projective boundary is (homeomorphic to) the disjoint union of projective spaces of the same dimension; see [6]. The latter fact relies on the observation that in the case of a nilpotent group the distance t .g 1 ; h 1 / is equal to the distance of the inverse elements t .g 1 ; h 1 / for all g 1 ; h 1 2 LG.…”