Hyperbolic groups with low-dimensional boundary

By Michael Kapovich, Bruce Kleiner. 2000

149 citations recorded by the source corpus.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/8a99f8deb354f493060a15386fac4f043916e400

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Rigidity of quasi-isometries for some hyperbolic buildings (2000, background): “Kleiner [KK] have produced examples of word hyperbolic groups where rigidity of quasi-isometries holds for purely topological reasons.”
Combinatorial modulus, the Combinatorial Loewner Property, and Coxeter groups (2010, background): “Since ∂Γ does not admit any local cut point – it is homeomorphic to the Menger curve, see [KK00] – Proposition 8.”
Quasisymmetric rigidity of square Sierpinski carpets (2011, background): “ are only nitely many distinct orbits of peripheral circles under the action of G, and hence of QS(S), on S. In this sense, Sis very symmetric. According to an open conjecture by Kapovich and Kleiner [KK], up to virtual isomorphism the groups Gas above are precisely the Gromov hyperbolic groups whos”
KLEINIAN GROUPS IN HIGHER DIMENSIONS (2007, background): “It was proved in [122] that this conjecture would follow either from the positive solution of the 3-dimensional Wall’s problem (Problem 4.”
Quasi-Actions on Trees II: Finite Depth Bass-Serre Trees (2004, unknown): “result does not hold in general for patterns of geodesics on hyperbolic surfaces. However, Kapovich and Kleiner show that pattern rigidity does hold when the geodesics c1,··· ,ck ﬁll the surface; see [KK00] Lemma 22. This is equivalent to our assumption on crossing graphs, and so we have: Theorem 7.”
Uniformization of Sierpiński carpets in the plane (2010, background): “try. More precisely, G should admit an action on a convex subset of hyperbolic 3-space H3 with non-empty totally geodesic boundary where the action is isometric, properly discontinuous, and cocompact [KK]. If Gadmits such an action on H3, then ∂ ∞Gcan be identiﬁed with 4 MARIO BONK a round Sierpin´s”
The Ergodic Theory of Hyperbolic Groups (2011, background): “In fact, a theorem of Kapovich–Kleiner [25] says that if G is a hyperbolic group which does not split over a finite or virtually cyclic subgroup, and if ∂∞G is 1dimensional (in the topological sense of dimension), then ∂∞G is homeomorphic to the circle, the Sierpinski carpet, or the Menger sponge.”