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14
The asymptotic geometry of negatively curved spaces: uniformization, geometrization and rigidity
 ICM Proceedings
, 2006
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EXISTENCE OF QUASIARCS
, 2008
"... Abstract. We show that doubling, linearly connected metric spaces are quasiarc connected. This gives a new and short proof of a theorem of Tukia. 1. ..."
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Cited by 6 (3 self)
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Abstract. We show that doubling, linearly connected metric spaces are quasiarc connected. This gives a new and short proof of a theorem of Tukia. 1.
Quasihyperbolic planes in relatively hyperbolic groups
, 2012
"... We show that any group that is hyperbolic relative to virtually nilpotent subgroups, and does not admit certain splittings, contains a quasiisometrically embedded copy of the hyperbolic plane. The specific embeddings we find remain quasiisometric embeddings when composed with the natural map f ..."
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Cited by 4 (3 self)
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We show that any group that is hyperbolic relative to virtually nilpotent subgroups, and does not admit certain splittings, contains a quasiisometrically embedded copy of the hyperbolic plane. The specific embeddings we find remain quasiisometric embeddings when composed with the natural map from the Cayley graph to the conedoff graph, as well as when composed with the quotient map to “almost every ” peripheral (Dehn) filling. We apply our theorem to study the same question for fundamental groups of 3manifolds. The proofs of these theorems involve quantitative geometric properties of the boundaries of relatively hyperbolic groups, such as linear connectedness. In particular, we prove a new existence result for quasiarcs that avoid obstacles.
SIMPLY CONNECTED HOMOGENEOUS CONTINUA ARE NOT SEPARATED BY ARCS
, 2006
"... Abstract. We show that locally connected, simply connected homogeneous continua are not separated by arcs. We ask several questions about homogeneous continua which are inspired by analogous questions in geometric group theory. 1. ..."
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Abstract. We show that locally connected, simply connected homogeneous continua are not separated by arcs. We ask several questions about homogeneous continua which are inspired by analogous questions in geometric group theory. 1.
QUASICIRCLES THROUGH PRESCRIBED POINTS
"... Abstract. We show that in an Lannularly linearly connected, Ndoubling, complete metric space, any n points lie on a λquasicircle, where λ depends only on L, N and n. This implies, for example, that if G is a hyperbolic group that does not split over any virtually cyclic subgroup, then any geodesi ..."
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Abstract. We show that in an Lannularly linearly connected, Ndoubling, complete metric space, any n points lie on a λquasicircle, where λ depends only on L, N and n. This implies, for example, that if G is a hyperbolic group that does not split over any virtually cyclic subgroup, then any geodesic line in G lies in a quasiisometrically embedded copy of H2. 1.
ON GROUPS WITH LINEAR SCI GROWTH
"... Abstract. We prove that the simple connectivity at infinity growth of sci hyperbolic groups and most nonuniform lattices is linear. Using the fact that the enddepth of finitely presented groups is linear we prove that the linear growth of simple connectivity at infinity is preserved under amalgama ..."
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Abstract. We prove that the simple connectivity at infinity growth of sci hyperbolic groups and most nonuniform lattices is linear. Using the fact that the enddepth of finitely presented groups is linear we prove that the linear growth of simple connectivity at infinity is preserved under amalgamated products over finitely generated oneended groups.