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Journal articles on the topic 'Relatively hyperbolic group'

Author: Grafiati

Published: 15 December 2023

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1

BOWDITCH, B. H. "RELATIVELY HYPERBOLIC GROUPS." International Journal of Algebra and Computation 22, no. 03 (May 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 (February 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 (December 3, 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 (June 30, 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 (August 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 (November 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 (March 26, 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 (September 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 (October 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 (March 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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11

Fukaya, Tomohiro, and Shin-ichi Oguni. "Coronae of relatively hyperbolic groups and coarse cohomologies." Journal of Topology and Analysis 08, no. 03 (June 8, 2016): 431–74. http://dx.doi.org/10.1142/s1793525316500151.

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We construct a corona of a relatively hyperbolic group by blowing-up all parabolic points of its Bowditch boundary. We relate the [Formula: see text]-homology of the corona with the [Formula: see text]-theory of the Roe algebra, via the coarse assembly map. We also establish a dual theory, that is, we relate the [Formula: see text]-theory of the corona with the [Formula: see text]-theory of the reduced stable Higson corona via the coarse co-assembly map. For that purpose, we formulate generalized coarse cohomology theories. As an application, we give an explicit computation of the [Formula: see text]-theory of the Roe-algebra and that of the reduced stable Higson corona of the fundamental groups of closed 3-dimensional manifolds and of pinched negatively curved complete Riemannian manifolds with finite volume.

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12

OSIN, DENIS V. "ELEMENTARY SUBGROUPS OF RELATIVELY HYPERBOLIC GROUPS AND BOUNDED GENERATION." International Journal of Algebra and Computation 16, no. 01 (February 2006): 99–118. http://dx.doi.org/10.1142/s0218196706002901.

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Let G be a group hyperbolic relative to a collection of subgroups {Hλ, λ ∈ Λ}. We say that a subgroup Q ≤ G is hyperbolically embedded into G, if G is hyperbolic relative to {Hλ, λ ∈ Λ} ∪ {Q}. In this paper we obtain a characterization of hyperbolically embedded subgroups. In particular, we show that if an element g ∈ G has infinite order and is not conjugate to an element of some Hλ, λ ∈ Λ, then the (unique) maximal elementary subgroup containing g is hyperbolically embedded into G. This allows us to prove that if G is boundedly generated, then G is elementary or Hλ = G for some λ ∈ Λ.

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13

Hruska, G. C., and Daniel T. Wise. "Finiteness properties of cubulated groups." Compositio Mathematica 150, no. 3 (March 2014): 453–506. http://dx.doi.org/10.1112/s0010437x13007112.

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AbstractWe give a generalized and self-contained account of Haglund–Paulin’s wallspaces and Sageev’s construction of the CAT(0) cube complex dual to a wallspace. We examine criteria on a wallspace leading to finiteness properties of its dual cube complex. Our discussion is aimed at readers wishing to apply these methods to produce actions of groups on cube complexes and understand their nature. We develop the wallspace ideas in a level of generality that facilitates their application. Our main result describes the structure of dual cube complexes arising from relatively hyperbolic groups. Let $H_1,\ldots, H_s$ be relatively quasiconvex codimension-1 subgroups of a group $G$ that is hyperbolic relative to $P_1, \ldots, P_r$. We prove that $G$ acts relatively cocompactly on the associated dual CAT(0) cube complex $C$. This generalizes Sageev’s result that $C$ is cocompact when $G$ is hyperbolic. When $P_1,\ldots, P_r$ are abelian, we show that the dual CAT(0) cube complex $C$ has a $G$-cocompact CAT(0) truncation.

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14

Diekert, Volker, Olga Kharlampovich, and Atefeh Mohajeri Moghaddam. "SLP compression for solutions of equations with constraints in free and hyperbolic groups." International Journal of Algebra and Computation 25, no. 01n02 (February 2015): 81–111. http://dx.doi.org/10.1142/s0218196715400056.

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The paper is a part of an ongoing program which aims to show that the problem of satisfiability of a system of equations in a free group (hyperbolic or even toral relatively hyperbolic group) is NP-complete. For that, we study compression of solutions with straight-line programs (SLPs) as suggested originally by Plandowski and Rytter in the context of a single word equation. We review some basic results on SLPs and give full proofs in order to keep this fundamental part of the program self-contained. Next we study systems of equations with constraints in free groups and more generally in free products of abelian groups. We show how to compress minimal solutions with extended Parikh-constraints. This type of constraints allows to express semi-linear conditions as e.g. alphabetic information. The result relies on some combinatorial analysis and has not been shown elsewhere. We show similar compression results for Boolean formula of equations over a torsion-free δ-hyperbolic group. The situation is much more delicate than in free groups. As byproduct we improve the estimation of the "capacity" constant used by Rips and Sela in their paper "Canonical representatives and equations in hyperbolic groups" from a double-exponential bound in δ to some single-exponential bound. The final section shows compression results for toral relatively hyperbolic groups using the work of Dahmani: We show that given a system of equations over a fixed toral relatively hyperbolic group, for every solution of length N there is an SLP for another solution such that the size of the SLP is bounded by some polynomial p(s + log N) where s is the size of the system.

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15

Ghosh, Pritam. "Relative hyperbolicity of free-by-cyclic extensions." Compositio Mathematica 159, no. 1 (January 2023): 153–83. http://dx.doi.org/10.1112/s0010437x22007813.

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Given a finitely generated free group $ {\mathbb {F} }$ of $\mathsf {rank}( {\mathbb {F} } )\geq 3$ , we show that the mapping torus of $\phi$ is (strongly) relatively hyperbolic if $\phi$ is exponentially growing. As a corollary of our work, we give a new proof of Brinkmann's theorem which proves that the mapping torus of an atoroidal outer automorphism is hyperbolic. We also give a new proof of the Bridson–Groves theorem that the mapping torus of a free group automorphism satisfies the quadratic isoperimetric inequality. Our work also solves a problem posed by Minasyan and Osin: the mapping torus of an outer automorphism is not virtually acylindrically hyperbolic if and only if $\phi$ has finite order.

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16

Dussaule, Matthieu, and Wenyuan Yang. "The Hausdorff dimension of the harmonic measure for relatively hyperbolic groups." Transactions of the American Mathematical Society, Series B 10, no. 23 (June 28, 2023): 766–806. http://dx.doi.org/10.1090/btran/145.

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The paper studies the Hausdorff dimension of harmonic measures on various boundaries of a relatively hyperbolic group which are associated with random walks driven by a probability measure with finite first moment. With respect to the Floyd metric and the shortcut metric, we prove that the Hausdorff dimension of the harmonic measure equals the ratio of the entropy and the drift of the random walk. If the group is infinitely-ended, the same dimension formula is obtained for the end boundary endowed with a visual metric. In addition, the Hausdorff dimension of the visual metric is identified with the growth rate of the word metric. These results are complemented by a characterization of doubling visual metrics for accessible infinitely-ended groups: the visual metrics on the end boundary is doubling if and only if the group is virtually free. Consequently, there are at least two different bi-Hölder classes (and thus quasi-symmetric classes) of visual metrics on the end boundary.

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17

Bogopolski, Oleg, and Kai-Uwe Bux. "From local to global conjugacy of subgroups of relatively hyperbolic groups." International Journal of Algebra and Computation 27, no. 03 (March 20, 2017): 299–314. http://dx.doi.org/10.1142/s021819671750014x.

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Suppose that a finitely generated group [Formula: see text] is hyperbolic relative to a collection of subgroups [Formula: see text]. Let [Formula: see text] be subgroups of [Formula: see text] such that [Formula: see text] is relatively quasiconvex with respect to [Formula: see text] and [Formula: see text] is not parabolic. Suppose that [Formula: see text] is elementwise conjugate into [Formula: see text]. Then there exists a finite index subgroup of [Formula: see text] which is conjugate into [Formula: see text]. The minimal length of the conjugator can be estimated. In the case, where [Formula: see text] is a limit group, it is sufficient to assume only that [Formula: see text] is a finitely generated and [Formula: see text] is an arbitrary subgroup of [Formula: see text].

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18

Einstein, Eduard, and Daniel Groves. "Relative cubulations and groups with a 2-sphere boundary." Compositio Mathematica 156, no. 4 (March 24, 2020): 862–67. http://dx.doi.org/10.1112/s0010437x20007095.

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We introduce a new kind of action of a relatively hyperbolic group on a $\text{CAT}(0)$ cube complex, called a relatively geometric action. We provide an application to characterize finite-volume Kleinian groups in terms of actions on cube complexes, analogous to the results of Markovic and Haïssinsky in the closed case.

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19

Martínez-Pedroza, Eduardo, and Piotr Przytycki. "DISMANTLABLE CLASSIFYING SPACE FOR THE FAMILY OF PARABOLIC SUBGROUPS OF A RELATIVELY HYPERBOLIC GROUP." Journal of the Institute of Mathematics of Jussieu 18, no. 2 (April 11, 2017): 329–45. http://dx.doi.org/10.1017/s147474801700010x.

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Let $G$ be a group hyperbolic relative to a finite collection of subgroups ${\mathcal{P}}$. Let ${\mathcal{F}}$ be the family of subgroups consisting of all the conjugates of subgroups in ${\mathcal{P}}$, all their subgroups, and all finite subgroups. Then there is a cocompact model for $E_{{\mathcal{F}}}G$. This result was known in the torsion-free case. In the presence of torsion, a new approach was necessary. Our method is to exploit the notion of dismantlability. A number of sample applications are discussed.

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20

Sisto, Alessandro. "Contracting elements and random walks." Journal für die reine und angewandte Mathematik (Crelles Journal) 2018, no. 742 (September 1, 2018): 79–114. http://dx.doi.org/10.1515/crelle-2015-0093.

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Abstract We define a new notion of contracting element of a group and we show that contracting elements coincide with hyperbolic elements in relatively hyperbolic groups, pseudo-Anosovs in mapping class groups, rank one isometries in groups acting properly on proper {\mathrm{CAT}(0)} spaces, elements acting hyperbolically on the Bass–Serre tree in graph manifold groups. We also define a related notion of weakly contracting element, and show that those coincide with hyperbolic elements in groups acting acylindrically on hyperbolic spaces and with iwips in {\mathrm{Out}(F_{n})} , {n\geq 3} . We show that each weakly contracting element is contained in a hyperbolically embedded elementary subgroup, which allows us to answer a problem in [16]. We prove that any simple random walk in a non-elementary finitely generated subgroup containing a (weakly) contracting element ends up in a non-(weakly-)contracting element with exponentially decaying probability.

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21

YANG, WEN-YUAN. "Growth tightness for groups with contracting elements." Mathematical Proceedings of the Cambridge Philosophical Society 157, no. 2 (July 30, 2014): 297–319. http://dx.doi.org/10.1017/s0305004114000322.

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AbstractWe establish growth tightness for a class of groups acting geometrically on a geodesic metric space and containing a contracting element. As a consequence, any group with non-trivial Floyd boundary are proven to be growth tight with respect to word metrics. In particular, all non-elementary relatively hyperbolic group are growth tight. This generalizes previous works of Arzhantseva-Lysenok and Sambusetti. Another interesting consequence is that CAT(0) groups with rank-1 elements are growth tight with respect to CAT(0)-metric.

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22

Kharlampovich, Olga, and Alexei Myasnikov. "Equations in Algebras." International Journal of Algebra and Computation 28, no. 08 (December 2018): 1517–33. http://dx.doi.org/10.1142/s0218196718400064.

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We show that the Diophantine problem (decidability of equations) is undecidable in free associative algebras over any field and in the group algebras over any field of a wide variety of torsion free groups, including toral relatively hyperbolic groups, right-angled Artin groups, commutative transitive groups, the fundamental groups of various graph groups, etc.

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23

HILDEN, HUGH M., MARIA TERESA LOZANO, and JOSÉ MARIA MONTESINOS-AMILIBIA. "UNIVERSAL 2-BRIDGE KNOT AND LINK ORBIFOLDS." Journal of Knot Theory and Its Ramifications 02, no. 02 (June 1993): 141–48. http://dx.doi.org/10.1142/s021821659300009x.

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Let (p/q, n) be the orbifold with cyclic isotropy of order n and with singular set the 2-bridge knot or link p/q where p and q are relatively prime numbers, q is odd, q is less than p, and q is not congruent to ±1 mod p (i.e. p/q is any non toroidal 2-bridge knot or link). We show that the orbifold fundamental group π1(p/q, n) is universal for n any multiple of 12. This means that if Γ is any such group, it can be thought of as a discrete group of hyperbolic isometries of hyperbolic 3-space ℍ3, and then, given any closed, oriented 3-manifold M, there exists a subgroup of finite index G of Γ such that M is homeomorphic to G\ℍ3. Since we have shown elsewhere that the group π1(5/3, 12) is an arithmetic group, it follows that there exists an orbifold, namely (5/3, 12), whose singular set is a knot, the figure eight, and whose fundamental group is both arithmetic and universal.

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24

Sardar, Pranab. "Packing subgroups in solvable groups." International Journal of Algebra and Computation 25, no. 05 (August 2015): 917–26. http://dx.doi.org/10.1142/s0218196715500253.

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We show that any subgroup of a (virtually) nilpotent-by-polycyclic group satisfies the bounded packing property of Hruska–Wise [Packing subgroups in relatively hyperbolic groups, Geom. Topol. 13 (2009) 1945–1988]. In particular, the same is true for all finitely generated subgroups of metabelian groups and linear solvable groups. However, we find an example of a finitely generated solvable group of derived length 3 which admits a finitely generated metabelian subgroup without the bounded packing property. In this example the subgroup is a retract also. Thus we obtain a negative answer to Problem 2.27 of the above paper. On the other hand, we show that polycyclic subgroups of solvable groups satisfy the bounded packing property.

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25

GOODMAN, OLIVER, and MICHAEL SHAPIRO. "ON A GENERALIZATION OF DEHN'S ALGORITHM." International Journal of Algebra and Computation 18, no. 07 (November 2008): 1137–77. http://dx.doi.org/10.1142/s0218196708004822.

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Viewing Dehn's algorithm as a rewriting system, we generalize to allow an alphabet containing letters which do not necessarily represent group elements. This extends the class of groups for which the algorithm solves the word problem to include finitely generated nilpotent groups, many relatively hyperbolic groups including geometrically finite groups and fundamental groups of certain geometrically decomposable 3-manifolds. The class has several nice closure properties. We also show that if a group has an infinite subgroup and one of exponential growth, and they commute, then it does not admit such an algorithm. We dub these Cannon's algorithms.

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26

Yang, Wen-yuan. "Statistically Convex-Cocompact Actions of Groups with Contracting Elements." International Mathematics Research Notices 2019, no. 23 (February 5, 2018): 7259–323. http://dx.doi.org/10.1093/imrn/rny001.

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Abstract This paper presents a study of the asymptotic geometry of groups with contracting elements, with emphasis on a subclass of statistically convex-cocompact (SCC) actions. The class of SCC actions includes relatively hyperbolic groups, CAT(0) groups with rank-1 elements, and mapping class groups acting on Teichmüller spaces, among others. We exploit an extension lemma to prove that a group with SCC actions contains large free sub-semigroups, has purely exponential growth, and contains a class of barrier-free sets with a growth-tight property. Our study produces new results and recovers existing ones for many interesting groups through a unified and elementary approach.

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Rourke, Shane O. "A combination theorem for affine tree-free groups." International Journal of Algebra and Computation 26, no. 07 (November 2016): 1283–321. http://dx.doi.org/10.1142/s0218196716500557.

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Let [Formula: see text] be an ordered abelian group. We show how a group admitting a free affine action without inversions on a [Formula: see text]-tree admits a natural graph of groups decomposition, where vertex groups inherit actions on [Formula: see text]-trees. We introduce a stronger condition (essential freeness) on an affine action and apply recent work of various authors to deduce that a finitely generated group admitting an essentially free affine action on a [Formula: see text]-tree is relatively hyperbolic with nilpotent parabolics, is locally relatively quasiconvex, and has solvable word, conjugacy and isomorphism problems. Conversely, given a graph of groups satisfying certain conditions, we show how an affine action of its fundamental group can be constructed. Specialising to the case of free affine actions, we obtain a large class of groups that have a free affine action on a [Formula: see text]-tree but that do not act freely by isometries on any [Formula: see text]-tree. We also give an example of a group that admits a free isometric action on a [Formula: see text]-tree but which is not residually nilpotent.

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28

YANG, WEN-YUAN. "Purely exponential growth of cusp-uniform actions." Ergodic Theory and Dynamical Systems 39, no. 3 (June 20, 2017): 795–831. http://dx.doi.org/10.1017/etds.2017.37.

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Suppose that a countable group$G$admits a cusp-uniform action on a hyperbolic space$(X,d)$such that$G$is of divergent type. The main result of the paper is characterizing the purely exponential growth type of the orbit growth function by a condition introduced by Dal’bo, Otal and Peigné [Séries de Poincaré des groupes géométriquement finis.Israel J. Math.118(3) (2000), 109–124]. For geometrically finite Cartan–Hadamard manifolds with pinched negative curvature, this condition ensures the finiteness of Bowen–Margulis–Sullivan measures. In this case, our result recovers a theorem of Roblin (in a coarse form). Our main tool is the Patterson–Sullivan measures on the Gromov boundary of$X$, and a variant of the Sullivan shadow lemma called the partial shadow lemma. This allows us to prove that the purely exponential growth of either cones, or partial cones or horoballs is also equivalent to the Dal’bo–Otal–Peigné condition. These results are used further in a paper by the present author [W. Yang, Patterson–Sullivan measures and growth of relatively hyperbolic groups.Preprint, 2013,arXiv:1308.6326].

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DELGADO, MANUEL, STUART MARGOLIS, and BENJAMIN STEINBERG. "COMBINATORIAL GROUP THEORY, INVERSE MONOIDS, AUTOMATA, AND GLOBAL SEMIGROUP THEORY." International Journal of Algebra and Computation 12, no. 01n02 (February 2002): 179–211. http://dx.doi.org/10.1142/s0218196702000924.

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This paper explores various connections between combinatorial group theory, semigroup theory, and formal language theory. Let G = <A|R> be a group presentation and ℬA, R its standard 2-complex. Suppose X is a 2-complex with a morphism to ℬA, R which restricts to an immersion on the 1-skeleton. Then we associate an inverse monoid to X which algebraically encodes topological properties of the morphism. Applications are given to separability properties of groups. We also associate an inverse monoid M(A, R) to the presentation <A|R> with the property that pointed subgraphs of covers of ℬA, R are classified by closed inverse submonoids of M(A, R). In particular, we obtain an inverse monoid theoretic condition for a subgroup to be quasiconvex allowing semigroup theoretic variants on the usual proofs that the intersection of such subgroups is quasiconvex and that such subgroups are finitely generated. Generalizations are given to non-geodesic combings. We also obtain a formal language theoretic equivalence to quasiconvexity which holds even for groups which are not hyperbolic. Finally, we illustrate some applications of separability properties of relatively free groups to finite semigroup theory. In particular, we can deduce the decidability of various semidirect and Mal/cev products of pseudovarieties of monoids with equational pseudovarieties of nilpotent groups and with the pseudovariety of metabelian groups.

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JI, RONGHUI, CRICHTON OGLE, and BOBBY RAMSEY. "${\mathcal{B}$-BOUNDED COHOMOLOGY AND APPLICATIONS." International Journal of Algebra and Computation 23, no. 01 (February 2013): 147–204. http://dx.doi.org/10.1142/s0218196713500057.

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A discrete group with word-length (G, L) is [Formula: see text]-isocohomological for a bounding classes [Formula: see text] if the comparison map from [Formula: see text]-bounded cohomology to ordinary cohomology (with coefficients in ℂ) is an isomorphism; it is strongly [Formula: see text]-isocohomological if the same is true with arbitrary coefficients. In this paper we establish some basic conditions guaranteeing strong [Formula: see text]-isocohomologicality. In particular, we show strong [Formula: see text]-isocohomologicality for an FP∞ group G if all of the weighted G-sensitive Dehn functions are [Formula: see text]-bounded. Such groups include all [Formula: see text]-asynchronously combable groups; moreover, the class of such groups is closed under constructions arising from groups acting on an acyclic complex. We also provide examples where the comparison map fails to be injective, as well as surjective, and give an example of a solvable group with quadratic first Dehn function, but exponential second Dehn function. Finally, a relative theory of [Formula: see text]-bounded cohomology of groups with respect to subgroups is introduced. Relative isocohomologicality is determined in terms of a new notion of relative Dehn functions and a relativeFP∞ property for groups with respect to a collection of subgroups. Applications for computing [Formula: see text]-bounded cohomology of groups are given in the context of relatively hyperbolic groups and developable complexes of groups.

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31

Szczepański, Andrzej. "Relatively hyperbolic groups." Michigan Mathematical Journal 45, no. 3 (December 1998): 611–18. http://dx.doi.org/10.1307/mmj/1030132303.

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32

Farb, B. "Relatively Hyperbolic Groups." Geometric And Functional Analysis 8, no. 5 (November 1, 1998): 810–40. http://dx.doi.org/10.1007/s000390050075.

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33

Mackay, John M., and Alessandro Sisto. "Quasi-hyperbolic planes in relatively hyperbolic groups." Annales Academiae Scientiarum Fennicae Mathematica 45, no. 1 (January 2020): 139–74. http://dx.doi.org/10.5186/aasfm.2020.4511.

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34

PAL, ABHIJIT, and SUMAN PAUL. "COMPLEX OF RELATIVELY HYPERBOLIC GROUPS." Glasgow Mathematical Journal 61, no. 03 (October 9, 2018): 657–72. http://dx.doi.org/10.1017/s0017089518000423.

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AbstractIn this paper, we prove a combination theorem for a complex of relatively hyperbolic groups. It is a generalization of Martin’s (Geom. Topology 18 (2014), 31–102) work for combination of hyperbolic groups over a finite MK-simplicial complex, where k ≤ 0.

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35

Xie, Xiangdong. "Growth of relatively hyperbolic groups." Proceedings of the American Mathematical Society 135, no. 3 (September 15, 2006): 695–704. http://dx.doi.org/10.1090/s0002-9939-06-08537-6.

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36

Reyes, Eduardo. "On cubulated relatively hyperbolic groups." Geometry & Topology 27, no. 2 (May 16, 2023): 575–640. http://dx.doi.org/10.2140/gt.2023.27.575.

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37

Bartels, A. "Coarse flow spaces for relatively hyperbolic groups." Compositio Mathematica 153, no. 4 (March 9, 2017): 745–79. http://dx.doi.org/10.1112/s0010437x16008216.

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We introduce coarse flow spaces for relatively hyperbolic groups and use them to verify a regularity condition for the action of relatively hyperbolic groups on their boundaries. As an application the Farrell–Jones conjecture for relatively hyperbolic groups can be reduced to the peripheral subgroups (up to index-2 overgroups in the $L$-theory case).

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38

Guirardel, Vincent, and Gilbert Levitt. "McCool groups of toral relatively hyperbolic groups." Algebraic & Geometric Topology 15, no. 6 (December 31, 2015): 3485–534. http://dx.doi.org/10.2140/agt.2015.15.3485.

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39

Hu, Yao, Tai-Hua Yan, and Feng-Wen Chen. "Energy and Environment Performance of Resource-Based Cities in China: A Non-Parametric Approach for Estimating Hyperbolic Distance Function." International Journal of Environmental Research and Public Health 17, no. 13 (July 3, 2020): 4795. http://dx.doi.org/10.3390/ijerph17134795.

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Scientific determination of energy and environmental efficiency and productivity is the key foundation of green development policy-making. The hyperbolic distance function (HDF) model can deal with both desirable output and undesirable output asymmetrically, and measure efficiency from the perspective of “increasing production and reducing pollution”. In this paper, a nonparametric linear estimation method of an HDF model including uncontrollable index and undesirable output is proposed. Under the framework of global reference, the changes of energy environmental efficiency and productivity and their factorization of 107 resource-based cities in China from 2003 to 2018 are calculated and analyzed. With the classification of resource-based cities by resource dependence (RD) and region, we discuss the feature in green development quality of those cities. The results show that: (1) On the whole, the average annual growth rate of energy and environmental productivity of resource-based cities in China is 2.6%, which is mainly due to technological changes. The backward of relative technological efficiency hinders the further growth of productivity, while the scale diseconomy is the main reason for the backward of relative technological efficiency. (2) For the classification of RD, the energy and environmental efficiency of the high-dependent group are significantly lower than the other two, and the growth of productivity of the medium-dependent group is the highest. (3) In terms of classification by region, the energy and environmental efficiency of the eastern region is the highest, and that of the middle and western regions is not as good as that of the eastern and northeastern regions. The middle region shows the situation of “middle collapse” in both static efficiency and dynamic productivity change, and the main reason for its low productivity growth is the retreat of relatively pure technical efficiency. This conclusion provides practical reference for the classification and implementation of regional energy and environmental policies.

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Osborne, Jeremy, and Wen-yuan Yang. "Statistical hyperbolicity of relatively hyperbolic groups." Algebraic & Geometric Topology 16, no. 4 (September 12, 2016): 2143–58. http://dx.doi.org/10.2140/agt.2016.16.2143.

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41

Hruska, G. Christopher, and Daniel T. Wise. "Packing subgroups in relatively hyperbolic groups." Geometry & Topology 13, no. 4 (April 21, 2009): 1945–88. http://dx.doi.org/10.2140/gt.2009.13.1945.

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42

Minasyan, A., and D. Osin. "Normal automorphisms of relatively hyperbolic groups." Transactions of the American Mathematical Society 362, no. 11 (November 1, 2010): 6079. http://dx.doi.org/10.1090/s0002-9947-2010-05067-6.

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43

Dadarlat, Marius, and Erik Guentner. "Uniform embeddability of relatively hyperbolic groups." Journal für die reine und angewandte Mathematik (Crelles Journal) 2007, no. 612 (January 1, 2007): 1–15. http://dx.doi.org/10.1515/crelle.2007.081.

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Osin, D. V. "Peripheral fillings of relatively hyperbolic groups." Electronic Research Announcements of the American Mathematical Society 12, no. 6 (April 28, 2006): 44–52. http://dx.doi.org/10.1090/s1079-6762-06-00159-4.

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45

Ramras, Daniel A., and Bobby W. Ramsey. "Extending properties to relatively hyperbolic groups." Kyoto Journal of Mathematics 59, no. 2 (June 2019): 343–56. http://dx.doi.org/10.1215/21562261-2018-0017.

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46

Ozawa, Narutaka. "Boundary amenability of relatively hyperbolic groups." Topology and its Applications 153, no. 14 (August 2006): 2624–30. http://dx.doi.org/10.1016/j.topol.2005.11.001.

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47

Osin, Denis V. "Peripheral fillings of relatively hyperbolic groups." Inventiones mathematicae 167, no. 2 (October 3, 2006): 295–326. http://dx.doi.org/10.1007/s00222-006-0012-3.

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48

Yang, Wen-yuan. "Limit sets of relatively hyperbolic groups." Geometriae Dedicata 156, no. 1 (February 9, 2011): 1–12. http://dx.doi.org/10.1007/s10711-011-9586-z.

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Groves, Daniel, and Jason Fox Manning. "Dehn filling in relatively hyperbolic groups." Israel Journal of Mathematics 168, no. 1 (September 19, 2008): 317–429. http://dx.doi.org/10.1007/s11856-008-1070-6.

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Dahmani, François. "Existential questions in (relatively) hyperbolic groups." Israel Journal of Mathematics 173, no. 1 (September 2009): 91–124. http://dx.doi.org/10.1007/s11856-009-0084-z.

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