Geodesic
LINEAR AND PROJECTIVE BOUNDARY OF NILPOTENT GROUPS
Glasgow mathematical journal, Tome 57 (2015) no. 3, pp. 591-632

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DOI

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. 
KRÖN, BERNHARD; LEHNERT, JÖRG; SEIFTER, NORBERT; TEUFL, ELMAR. LINEAR AND PROJECTIVE BOUNDARY OF NILPOTENT GROUPS. Glasgow mathematical journal, Tome 57 (2015) no. 3, pp. 591-632. doi: 10.1017/S0017089514000512
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     title = {LINEAR {AND} {PROJECTIVE} {BOUNDARY} {OF} {NILPOTENT} {GROUPS}},
     journal = {Glasgow mathematical journal},
     pages = {591--632},
     year = {2015},
     volume = {57},
     number = {3},
     doi = {10.1017/S0017089514000512},
     url = {http://geodesic.mathdoc.fr/articles/10.1017/S0017089514000512/}
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