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Relevant bibliographies by topics / Relatively hyperbolic group

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Journal articles

Academic literature on the topic 'Relatively hyperbolic group'

Author: Grafiati

Published: 15 December 2023

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Journal articles on the topic "Relatively hyperbolic group"

1

BOWDITCH, B. H. "RELATIVELY HYPERBOLIC GROUPS." International Journal of Algebra and Computation 22, no. 03 (2012): 1250016. http://dx.doi.org/10.1142/s0218196712500166.

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In this paper we develop some of the foundations of the theory of relatively hyperbolic groups as originally formulated by Gromov. We prove the equivalence of two definitions of this notion. One is essentially that of a group admitting a properly discontinuous geometrically finite action on a proper hyperbolic space, that is, such that every limit point is either a conical limit point or a bounded parabolic point. The other is that of a group which admits a cofinite action on a connected fine hyperbolic graph. We define a graph to be "fine" if there are only finitely many circuits a given length containing any given edge, and we develop some of the properties of this notion. We show how a relatively hyperbolic group can be assumed to act on a proper hyperbolic space of a particular geometric form. We define the boundary of a relatively hyperbolic group, and show that the limit set of any geometrically finite action of the group is equivariantly homeomorphic to this boundary. This generalizes a result of Tukia for geometrically finite kleinian groups. We also describe when the boundary is connected.

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2

BELEGRADEK, IGOR, and ANDRZEJ SZCZEPAŃSKI. "ENDOMORPHISMS OF RELATIVELY HYPERBOLIC GROUPS." International Journal of Algebra and Computation 18, no. 01 (2008): 97–110. http://dx.doi.org/10.1142/s0218196708004305.

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We generalize some results of Paulin and Rips-Sela on endomorphisms of hyperbolic groups to relatively hyperbolic groups, and in particular prove the following. • If G is a nonelementary relatively hyperbolic group with slender parabolic subgroups, and either G is not co-Hopfian or Out (G) is infinite, then G splits over a slender group. • If H is a nonparabolic subgroup of a relatively hyperbolic group, and if any isometric H-action on an ℝ-tree is trivial, then H is Hopfian. • If G is a nonelementary relatively hyperbolic group whose peripheral subgroups are finitely generated, then G has a nonelementary relatively hyperbolic quotient that is Hopfian. • Any finitely presented group is isomorphic to a finite index subgroup of Out (H) for some group H with Kazhdan property (T). (This sharpens a result of Ollivier–Wise).

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3

Mihalik, M., and E. Swenson. "Relatively hyperbolic groups with semistable fundamental group at infinity." Journal of Topology 14, no. 1 (2020): 39–61. http://dx.doi.org/10.1112/topo.12178.

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4

Gerasimov, Victor, and Leonid Potyagailo. "Similar Relatively Hyperbolic Actions of a Group." International Mathematics Research Notices 2016, no. 7 (2015): 2068–103. http://dx.doi.org/10.1093/imrn/rnv170.

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5

Bumagin, Inna. "Time complexity of the conjugacy problem in relatively hyperbolic groups." International Journal of Algebra and Computation 25, no. 05 (2015): 689–723. http://dx.doi.org/10.1142/s0218196715500162.

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If u and v are two conjugate elements of a hyperbolic group then the length of a shortest conjugating element for u and v can be bounded by a linear function of the sum of their lengths, as was proved by Lysenok in [Some algorithmic properties of hyperbolic groups, Izv. Akad. Nauk SSSR Ser. Mat. 53(4) (1989) 814–832, 912]. Bridson and Haefliger showed in [Metrics Spaces of Non-Positive Curvature (Springer-Verlag, Berlin, 1999)] that in a hyperbolic group the conjugacy problem can be solved in polynomial time. We extend these results to relatively hyperbolic groups. In particular, we show that both the conjugacy problem and the conjugacy search problem can be solved in polynomial time in a relatively hyperbolic group, whenever the corresponding problem can be solved in polynomial time in each parabolic subgroup. We also prove that if u and v are two conjugate hyperbolic elements of a relatively hyperbolic group then the length of a shortest conjugating element for u and v is linear in terms of their lengths.

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6

TRAN, HUNG CONG. "RELATIONS BETWEEN VARIOUS BOUNDARIES OF RELATIVELY HYPERBOLIC GROUPS." International Journal of Algebra and Computation 23, no. 07 (2013): 1551–72. http://dx.doi.org/10.1142/s0218196713500367.

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Suppose a group G is relatively hyperbolic with respect to a collection ℙ of its subgroups and also acts properly, cocompactly on a CAT(0) (or δ-hyperbolic) space X. The relatively hyperbolic structure provides a relative boundary ∂(G, ℙ). The CAT(0) structure provides a different boundary at infinity ∂X. In this paper, we examine the connection between these two spaces at infinity. In particular, we show that ∂(G, ℙ) is G-equivariantly homeomorphic to the space obtained from ∂X by identifying the peripheral limit points of the same type.

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7

Fujiwara, Koji. "Asymptotically isometric metrics on relatively hyperbolic groups and marked length spectrum." Journal of Topology and Analysis 07, no. 02 (2015): 345–59. http://dx.doi.org/10.1142/s1793525315500132.

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We prove asymptotically isometric, coarsely geodesic metrics on a toral relatively hyperbolic group are coarsely equal. The theorem applies to all lattices in SO (n, 1). This partly verifies a conjecture by Margulis. In the case of hyperbolic groups/spaces, our result generalizes a theorem by Furman and a theorem by Krat. We discuss an application to the isospectral problem for the length spectrum of Riemannian manifolds. The positive answer to this problem has been known for several cases. Most of them have hyperbolic fundamental groups. We do not solve the isospectral problem in the original sense, but prove the universal covers are (1, C)-quasi-isometric if the fundamental group is a toral relatively hyperbolic group.

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8

Dahmani, François, and Vincent Guirardel. "Recognizing a relatively hyperbolic group by its Dehn fillings." Duke Mathematical Journal 167, no. 12 (2018): 2189–241. http://dx.doi.org/10.1215/00127094-2018-0014.

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9

DAHMANI, FRANÇOIS. "PARABOLIC GROUPS ACTING ON ONE-DIMENSIONAL COMPACT SPACES." International Journal of Algebra and Computation 15, no. 05n06 (2005): 893–906. http://dx.doi.org/10.1142/s0218196705002530.

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Given a class of compact spaces, we ask which groups can be maximal parabolic subgroups of a relatively hyperbolic group whose boundary is in the class. We investigate the class of one-dimensional connected boundaries. We get that any non-torsion infinite finitely-generated group is a maximal parabolic subgroup of some relatively hyperbolic group with connected one-dimensional boundary without global cut point. For boundaries homeomorphic to a Sierpinski carpet or a 2-sphere, the only maximal parabolic subgroups allowed are virtual surface groups (hyperbolic, or virtually ℤ + ℤ).

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10

FUKAYA, TOMOHIRO, and SHIN-ICHI OGUNI. "THE COARSE BAUM–CONNES CONJECTURE FOR RELATIVELY HYPERBOLIC GROUPS." Journal of Topology and Analysis 04, no. 01 (2012): 99–113. http://dx.doi.org/10.1142/s1793525312500021.

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We study a group which is hyperbolic relative to a finite family of infinite subgroups. We show that the group satisfies the coarse Baum–Connes conjecture if each subgroup belonging to the family satisfies the coarse Baum–Connes conjecture and admits a finite universal space for proper actions. If the group is torsion-free, then it satisfies the analytic Novikov conjecture.

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