Relatively hyperbolic groups: Intrinsic geometry, algebraic properties, and algorithmic problems

By Denis V. Osin. 2004

498 citations recorded by the source corpus.

We suggest a new approach to the study of relatively hyperbolic groups based on relative isoperimetric inequalities. Various geometric, algebraic, and algorithmic properties are discussed.

Subfields: Applied Mathematics, Computer Vision and Pattern Recognition, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/a5d85b6c6977da4fada8c2613519b94b778bb317

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In Applied Mathematics

In Computer Vision and Pattern Recognition

In Geometry and Topology

Citations in the corpus

Boundary amenability of relatively hyperbolic groups (2005, unknown): “[AR][Ka] for simpler proofs.) In this paper, we study the case of relatively hyperbolic groups. The notion of relative hyperbolicity was proposed by Gromov [Gr1] and has been developed in [Bo][DS][Fa][Os1], just to name a few. Let G be a group generated by a ﬁnite symmetric subset S and let {Hi} be ”
Asymptotic Dimension (2007, background): “Osin showed in [78] that whenG is generated by a finite set in the ordinary sense and finitely presented relative to a collection of subgroups {Hλ}λ∈Λ then Λ is known to be finite and the subgroups Hλ are known to be finitely generated.”
Infinitely presented graphical small cancellation groups are acylindrically hyperbolic (2014, background): “yperbolic. We explain how the proofs from Section4.2adapt to the case of free products. Since in the case that is nite, we obtain a relatively hyperbolic group, we start out with denitions following [Osi06b]. Given a set S, M(S) denotes the free monoid on StS 1. Denition 4.18. Let Gbe a group and fG”
Groups acting on tree-graded spaces and splittings of relatively hyperbolic groups☆ (2006, background): “rtually cyclic or K is a parabolic subgroup. Proof. Suppose that Khas a central element zof inﬁnite order. If no power of zis in a parabolic subgroup then the normalizer of hzi is virtually cyclic by [Os], so Kis virtually cyclic. If zn is inside a maximal parabolic subgroup Hthen for every k∈ K, kH”
Hereditary conjugacy separability of right angled Artin groups and its applications (2009, methodology): “We refer the reader to Osin’s monograph [ 50 ] for the denition and basic properties of relatively hyperbolic groups.”
Residual finiteness, QCERF, and fillings of hyperbolic groups (2008, background): “rbolic ﬁlling G′ = G(N1,...,Nm) and Q as in Theorem 2.1 with |Ni|P i ≥ B for all i the space X(G′,Q,S) is δ-hyperbolic . Proof. By [7, Theorem 3.25], G is hyperbolic relative to P. By the Appendix of [13] this means that the relative Dehn function of G with respect to P is linear. Let C be the const”
Jsj Decompositions of Groups (2016, background): “containshyperbolicelements,thereis auniqueminimalinvariantsubtree T(G),andthequotient T(G)=Gisaﬁnitegraph. RecallthatTisrelativetoHifeveryH iisellipticinT. Wenowconsiderrelativeﬁnitepresentation(see[Osi06]). Notethat,if isarelative ﬁnitegeneratingset,thenthenaturalmorphismF( )H 1 H p!Gisanepimorphis”
Thick metric spaces, relative hyperbolicity, and quasi-isometric rigidity (2005, methodology): “The theory of relatively hyperbolic groups was developed by Farb [40], then further developed in [17,28,74,88], and [34].”
Asymptotic Geometry of the Mapping Class Group and Teichmuller Space (2005, background): “Many people have since studied another equivalent versions of relative hyperbolicity which was formulated by Gromov [22] and then elaborated on by Bowditch, Osin, Szczepa´ nski, and others [10; 48 ; 51].”
Finiteness properties of cubulated groups (2012, background): “relative to the peripheral subgroups if the action of G as a geometrically ﬁnite convergence group restricts to a geometrically ﬁnite convergence group action of H on its limit set. Subsequently Osin [Osi06] gave the following alternate formulation of relative quasiconvexity. In [Hru10], Hruska show”