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The local-to-global property for Morse quasi-geodesics

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Pith's one-line read The paper proves that Morse quasi-geodesics satisfy a local-to-global principle in a wide class of groups and spaces, including mapping class groups, CAT(0) spaces, and closed 3-manifold groups.

desk verdict A new local-to-global property for Morse quasi-geodesics, proved for a broad class of groups; the relative hyperbolicity proof has one compressed step that is more an exposition gap than a mathematical flaw. read the letter →

arxiv 1908.11292 v3 pith:EFKAXSE2 submitted 2019-08-29 math.GR math.GT

Jacob Russell , Davide Spriano , Hung Cong Tran This is my paper · ORCID
classification math.GRmath.GT MSC 20F6520F6757M07
keywords Morsequasi-geodesicslocal-to-globalpropertyrelativelyhyperbolicgroupsstablesubgroupsmappingclassgroupCAT(0)spaces3-manifolddeeppoints
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper establishes a local-to-global principle for Morse quasi-geodesics: if a path in a geodesic metric space has all of its subpaths of some uniform length as Morse quasi-geodesics, then the whole path is a Morse quasi-geodesic. It proves that the principle holds for CAT(0) spaces, mapping class groups, Teichmuller space, graph products of hyperbolic groups, virtually solvable groups, and fundamental groups of closed 3-manifolds. The proof's engine is a relative hyperbolicity inheritance theorem: a space hyperbolic relative to peripheral subsets inherits the property from those subsets. The argument introduces a "deep points" decomposition of local quasi-geodesics into pieces that alternately hug a peripheral subset and avoid all peripherals. If right, a suite of hyperbolic-space theorems, including stable subgroup combination theorems, discreteness of Morse translation lengths, and a local criterion for hyperbolicity, transfer to all these groups.

What carries the argument

The load-bearing mechanism is a theory of deep points for local quasi-geodesics in relatively hyperbolic spaces, extending Hruska's deep points for geodesics. A point in the domain is $P$-deep if witnesses on both sides run within a fixed neighborhood of a peripheral $P$ for a nontrivial length of time; deep points for different peripherals cannot overlap, so a local quasi-geodesic decomposes as $\sigma_0 * \alpha_1 * \sigma_1 * \cdots * \alpha_n * \sigma_n$. Each $\alpha_i$ is a long subsegment running near a single peripheral $P_i$, and each $\sigma_i$ has uniformly bounded projections to every peripheral, hence is a Morse quasi-geodesic. When each peripheral has the Morse local-to-global property, each deep $\alpha_i$ is likewise a global Morse quasi-geodesic, via the claimed inheritance of the property by thick neighborhoods. A linear-ordering argument for the relevant peripherals then forces any quasi-geodesic with the same endpoints to pass close to all the $\alpha_i$, so the whole local path is Morse.

What would settle it

Check the transfer in Corollary 5.23 directly: take a peripheral subset $P$ with the $\Phi$-Morse local-to-global property and ask whether every uniform thickening $N_{rR}(P)$ with the induced metric has a $\Psi$-Morse local-to-global property with $\Psi$ depending only on $\Phi$, $\lambda$, and $\epsilon$. A concrete counterexample, or a local Morse quasi-geodesic in a relatively hyperbolic space whose deep subsegment stays in such a thickening but is not a global Morse quasi-geodesic, would settle that the proof of Theorem 5.1 fails at this step.

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Extended reading notes

Core claim

The central claim is Theorem 5.1: if a geodesic metric space $X$ is hyperbolic relative to a collection $\mathcal{P}$ of peripheral subsets and every $P\in\mathcal{P}$ has the $\Phi$-Morse local-to-global property, then $X$ has the $\Psi$-Morse local-to-global property. A local Morse quasi-geodesic is a map whose subsegments of parametrized length at most $L$ are all $M$-Morse quasi-geodesics, and the property says that for every $M,\lambda,\epsilon$ there is a scale $L$ such that every such local path is a global Morse quasi-geodesic. The authors claim the inheritance is uniform: the constants for $X$ depend only on the constants for the peripherals. Together with known cases, this yields the broad list of examples in Theorem D: CAT(0) spaces, mapping class groups, Teichmuller space, graph products of hyperbolic groups, virtually solvable groups, and fundamental groups of closed 3-manifolds all have the Morse local-to-global property.

Load-bearing premise

The load-bearing premise is that the Morse local-to-global property automatically passes from each peripheral subset to every uniformly thick neighborhood of it with the induced metric; the proof uses this transfer without proving it.

Editorial extensions

If this is right

  • In any finitely generated Morse local-to-global group, stable subgroups satisfy Gitik-style combination theorems: under a short-element intersection condition, the subgroup they generate is their amalgamated free product and is itself stable (Theorem G).
  • In the mapping class group this yields combination theorems for convex cocompact subgroups, since stable subgroups there are exactly the convex cocompact ones (Theorem M).
  • Every infinite normal subgroup of a Morse local-to-global group that contains a Morse element must itself contain a Morse element (Corollary 3.6).
  • For each fixed Morse gauge, the set of translation lengths of $M$-Morse conjugacy classes is a discrete set of rational numbers (Theorem 3.12).
  • A geodesic metric space with the property that all sufficiently large local triangles are uniformly slim is globally hyperbolic (Theorem 3.15); in particular the universal covers of closed 3-manifolds and CAT(0) spaces pass this local check.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • One implicit consequence is that any group hyperbolic relative to Morse limited subgroups is Morse local-to-global, so the theorem automatically covers many groups built from unconstricted pieces, widening the class beyond the listed examples.
  • The deep-point decomposition may give a tool for transferring coarse contracting or divergence properties from peripherals to the ambient space, which could address open questions about whether Morse quasi-geodesics in Morse local-to-global spaces have stronger contracting or divergence behavior.
  • The unproved thick-neighborhood transfer in Corollary 5.23 is a testable gap: if a peripheral $P$ has the property but a uniform thickening $N_{rR}(P)$ with the induced metric does not, then the inheritance theorem would need a different argument and might fail.
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Editorial analysis

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Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper introduces a local-to-global property for Morse quasi-geodesics in metric spaces and proves that a wide range of groups and spaces satisfy it: CAT(0) spaces, mapping class groups and Teichmüller spaces, graph products of hyperbolic groups, virtually solvable groups, and fundamental groups of closed 3-manifolds. The main technical contribution is a relatively hyperbolic inheritance theorem (Theorem 5.1): if a geodesic metric space is hyperbolic relative to peripheral subsets each having the Morse local-to-global property, then the ambient space has the property. To prove this, the authors develop a theory of deep points for local quasi-geodesics, decomposing a local quasi-geodesic into alternating deep subsegments running close to peripherals and shallow subsegments with uniformly bounded projections. The paper also derives consequences of the property, including stable-subgroup combination theorems, discreteness of translation lengths for Morse conjugacy classes with a fixed gauge, and a Cartan–Hadamard-type local criterion for hyperbolicity.

Significance. If the relatively hyperbolic proof is completed, this is a substantial contribution. The Morse local-to-global property is a natural and useful strengthening of the behavior of Morse quasi-geodesics, and the paper shows it holds for many of the standard groups and spaces in geometric group theory. The deep-points machinery for local quasi-geodesics in relatively hyperbolic spaces is a genuinely new tool that likely has independent applications. The consequences are clean and broad: the combination theorems generalize results of Gitik from hyperbolic groups to stable subgroups, and the applications to mapping class groups and 3-manifold groups are valuable. The paper is careful in setting up uniform gauges and in separating the different ingredients of the proof, and the claimed results are consistent and plausible.

major comments (3)
  1. [§5.4, Corollary 5.23] The proof asserts without proof that if a peripheral subset P_i has the Φ–Morse local-to-global property, then a uniform neighborhood N_{rR}(P_i), equipped with the induced metric, has the Ψ–Morse local-to-global property. This transfer is not a formal consequence of the definition: the induced metric on a neighborhood can be very different from the ambient metric, and a subsegment of α_i that is a local Morse quasi-geodesic in X need not be a local Morse quasi-geodesic in N_{rR}(P_i) with the induced metric. Since Corollary 5.23 is used in Lemma 5.33 to conclude that every deep subsegment α_i is a global Morse quasi-geodesic in X, this is a load-bearing step. The authors should provide a proof or a precise citation establishing the transfer, with constants depending only on Φ, λ, and ε, uniformly over the family of peripherals.
  2. [§5.4, Corollary 5.23, final sentence] Even granting the transfer just discussed, the assertion that 'The distance formula (Theorem 5.7) and Lemma 5.5 then imply each α_i is an (N;k,c)–Morse quasi-geodesic in X' is not justified by the cited results. If β is a quasi-geodesic in X with endpoints on α_i, Lemma 5.5 only guarantees that β lies in N_{r' rR}(P_i) for some larger radius r' depending on the quasi-geodesic constants of β. But Corollary 5.23 establishes Morse control of α_i only inside N_{rR}(P_i), not inside the larger neighborhood. The distance formula cannot by itself bridge this gap without an additional argument showing that the relevant part of β is in fact controlled by α_i. Since the conclusion that each α_i is a global Morse quasi-geodesic in X is essential for Lemma 5.33 and hence for Theorem 5.1, this needs a complete proof.
  3. [§3.1, Theorem 3.1] In the proofs of parts (1) and (2), the verification that the concatenated path γ is a local quasi-geodesic is delegated to Gitik's arguments with the statement that 'this argument only uses' certain facts that remain true in the present setting. This is a substantial import of an external proof into a setting involving the Morse local-to-global property, and the specific quasi-geodesic constants (3, ε) and (6, ε) are claimed without reproducing the counting or minimality steps. The reader cannot easily verify that the adapted argument yields exactly those constants. Since Theorem 3.1 is one of the main advertised applications of the paper, the proof should either reproduce the relevant Gitik argument or give a precise reference and state explicitly the lemma being adapted.
minor comments (4)
  1. [§5.3, Corollary 5.22] The statement says 'if σ0 ˚ α1 ˚ σ1 ˚ ... ˚ αn ˚ σn is the B–relevant decomposition of an pL2;λ,ǫq–quasi-geodesic', but it should read 'pL2;λ,ǫq–local quasi-geodesic'.
  2. [§3.1, proof of Theorem 3.1(2)] In the sentence 'Once we show that γ is an pL;M1;6,cq–local Morse quasi-geodesic', the constant c should presumably be ε, consistent with the rest of the proof.
  3. [§4.1, Example 4.10(1)] The reference [Fin] is cited informally as 'Finks the author'; this should be replaced with the full name of the author and a complete bibliographic entry.
  4. [§5.1] The symbol R is used both for the neighborhood radius in Lemma 5.5 and for the constant R(λ, ε) in Lemma 5.6; although the usage is locally clear, a remark or notational distinction would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the relatively hyperbolic inheritance uses the peripheral local-to-global property as a hypothesis, not as a renamed conclusion.

full rationale

The paper's central derivation, Theorem 5.1, is not circular. The Morse local-to-global property (Definition B) is defined independently of the relatively hyperbolic theorem, and the proof of Theorem 5.1 supplies a deep-points decomposition (Section 5.4), bounded-projection reduction (Corollary 5.12), and a linear-ordering argument (Proposition 5.30) that turn the peripheral hypothesis into a global Morse conclusion. The most delicate step, Corollary 5.23, asserts without proof that N_{rR}(P_i) with the induced metric inherits the Ψ-Morse local-to-global property from P_i and that the relevant subsegments α_i are then Morse in X; this is a genuine correctness gap, but it is not circular because it is an unproved transfer lemma, not an identity or a fitted parameter renamed as a prediction. The applications in Sections 3 and 4 rely on external published results (e.g., Charney–Sultan [CS15], Abbott–Behrstock–Durham [ABD21], Tran [Tra19], Durham–Taylor [DT15]) whose assumptions are stated independently of the present target claims. Although [ABD21] has an appendix coauthored by Russell, its Morse-detectability theorem is a published external result with hypotheses (bounded domain dichotomy) that do not include the Morse local-to-global property, so this self-citation is not load-bearing in a circular sense. No definitional circularity, fitted-input prediction, or uniqueness-imported-from-authors pattern is present.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

This ledger is empty of free parameters and invented entities. The paper relies on a substantial body of established geometric group theory; the entries above are the background theorems whose truth the central argument takes as given. The only paper-specific premise is the inherited Morse local-to-global property for neighborhoods of peripherals, which is stated but not proved.

assumptions (7)
  • standard math Hierarchically hyperbolic spaces with the bounded domain dichotomy are Morse detectable (ABD21, Corollary 6.2 and Theorem 7.2).
    Invoked in Theorem 4.20 to prove mapping class groups, Teichmuller spaces, and graph products of hyperbolic groups have the Morse local-to-global property via Theorem 4.18.
  • standard math Stable subgroups are finitely generated, hyperbolic, undistorted, have finite width, and are finite index in their commensurator (DT15, Tra19, AMST19).
    Used throughout Section 3, especially Theorems 2.18 and Proposition 2.17, for the combination theorems and corollaries.
  • standard math The distance formula, bounded geodesic image, projections onto peripherals, and linear quasiconvexity hold in relatively hyperbolic spaces (Sis13, BHS19, DS05).
    These are the core tools of Section 5 and are taken as established results.
  • standard math Morse geodesics in CAT(0) spaces are exactly D-contracting geodesics (CS15, Theorem 2.9).
    Used in Theorem 4.16 to show CAT(0) spaces have the Morse local-to-global property.
  • standard math A geodesic metric space is hyperbolic if and only if local quasi-geodesics of sufficiently large scale are global quasi-geodesics (Gro87, Proposition 7.2.E).
    Used in Theorem 4.18 and in the proof of Proposition 5.13.
  • standard math Closed 3-manifolds decompose by prime decomposition with factors that are virtually solvable or hierarchically hyperbolic (Perelman geometrization, BHS19).
    Used in Corollary 5.2 to extend the Morse local-to-global property to all closed 3-manifold groups.
  • ad hoc to paper A uniform neighborhood N_{rR}(P_i) of a peripheral subset, with induced metric, has the Morse local-to-global property whenever P_i does.
    Stated in the proof of Corollary 5.23 without proof; probably follows from quasi-isometry invariance, but the uniformity over peripherals is not shown. This is the weakest premise in the relative hyperbolic argument.

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Pith. "Pith review of The local-to-global property for Morse quasi-geodesics." pith.science (2026). https://pith.science/paper/EFKAXSE2

@misc{pith2026190811292,
  author       = {Pith},
  title        = {Pith review of: The local-to-global property for Morse quasi-geodesics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EFKAXSE2}},
  note         = {Machine review of arXiv:1908.11292}
}
read the original abstract

We show the mapping class group, CAT(0) groups, the fundamental groups of closed 3-manifolds, and certain relatively hyperbolic groups have a local-to-global property for Morse quasi-geodesics. This allows us to generalize combination theorems of Gitik for quasiconvex subgroups of hyperbolic groups to the stable subgroups of these groups. In the case of the mapping class group, this gives combination theorems for convex cocompact subgroups. We show a number of additional consequences of this local-to-global property, including a Cartan-Hadamard type theorem for detecting hyperbolicity locally and discreteness of translation length of conjugacy classes of Morse elements with a fixed gauge. To prove the relatively hyperbolic case, we develop a theory of deep points for local quasi-geodesics in relatively hyperbolic spaces, extending work of Hruska.

Figures

Figures reproduced from arXiv: 1908.11292 by the authors.

Figure 1
Figure 1. Defining graph Γ for the right-angled Coxeter group in Example 3.4 from Theorem 3.1.(1). We produce the infinite family of subgroups by inductively separating Pi and Qi from the short elements of Pi´1 and Qi´1 that are not contained in P X Q. Corollary 3.3 allows us to produce a plethora of examples where our combination theorem applies with a non-trivial intersection between the subgroups. We demonstrate an explici… view at source ↗

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Works this paper leans on

46 extracted references · 29 canonical work pages

  1. [1]

    Carolyn Abbott, Jason Behrstock, and Matthew G. Durham. Largest acylindrical actions and stability in hierarchically hyperbolic groups. W ith an appendix by D aniel B erlyne and J acob R ussell. Trans. Amer. Math. Soc. Ser. B , 8:66--104, 2021

    work page 2021

  2. [2]

    Arzhantseva, Christopher H

    Goulnara N. Arzhantseva, Christopher H. Cashen, Dominik Gruber, and David Hume. Characterizations of M orse quasi-geodesics via superlinear divergence and sublinear contraction. Doc. Math. , 22:1193--1224, 2017

    work page 2017

  3. [3]

    Arzhantseva, Christopher H

    Goulnara N. Arzhantseva, Christopher H. Cashen, and Jing Tao. Growth tight actions. Pacific J. Math. , 278(1):1--49, 2015

    2015

  4. [4]

    Abbott and Fran c ois Dahmani

    Carolyn R. Abbott and Fran c ois Dahmani. Property P_ naive for acylindrically hyperbolic groups. Math. Z. , 291(1-2):555--568, 2019

    work page 2019

  5. [5]

    Tarik Aougab, Matthew G Durham, and Samuel J. Taylor. Pulling back stability with applications to Out (F_n) and relatively hyperbolic groups. J. Lond. Math. Soc. (2) , 96(3):565--583, 2017

    work page 2017

  6. [6]

    The virtual H aken conjecture

    Ian Agol. The virtual H aken conjecture. Doc. Math. , 18:1045--1087, 2013. With an appendix by Agol, Daniel Groves, and Jason Manning

    2013

  7. [7]

    Yago Antol\' n, Mahan Mj, Alessandro Sisto, and Samuel J. Taylor. Intersection properties of stable subgroups and bounded cohomology. Indiana Univ. Math. J. , 68(1):179--199, 2019

    work page 2019

  8. [8]

    Arzhantseva

    Goulnara N. Arzhantseva. On quasiconvex subgroups of word hyperbolic groups. Geom. Dedicata , 87(1-3):191--208, 2001

    2001

Show all 46 references
  1. [9]

    Divergence, thick groups, and short conjugators

    Jason Behrstock and Cornelia Dru t u. Divergence, thick groups, and short conjugators. Illinois J. Math. , 58(4):939--980, 2014

    2014

  2. [10]

    Bridson and Andr \'e Haefliger

    Martin R. Bridson and Andr \'e Haefliger. Metric spaces of non-positive curvature , volume 319 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] . Springer-Verlag, Berlin, 1999

    1999

  3. [11]

    Hagen, and Alessandro Sisto

    Jason Behrstock, Mark F. Hagen, and Alessandro Sisto. Hierarchically hyperbolic spaces II : C ombination theorems and the distance formula . Pacific J. Math. , 299:257--338, 2019

    2019

  4. [12]

    Bowditch

    Brian H. Bowditch. Notes on G romov's hyperbolicity criterion for path-metric spaces. In Group theory from a geometrical viewpoint ( T rieste, 1990) , pages 64--167. World Sci. Publ., River Edge, NJ, 1991

    1990

  5. [13]

    Bowditch

    Brian H. Bowditch. Tight geodesics in the curve complex. Invent. Math. , 171(2):281--300, 2008

    2008

  6. [14]

    James W. Cannon. The combinatorial structure of cocompact discrete hyperbolic groups. Geom. Dedicata , 16(2):123--148, 1984

    1984

  7. [15]

    Quasi- M obius homeomorphisms of M orse boundaries

    Ruth Charney, Matthew Cordes, and Devin Murray. Quasi- M obius homeomorphisms of M orse boundaries. Bull. Lond. Math. Soc. , 51(3):501--515, 2019

    2019

  8. [16]

    Stability and the M orse boundary

    Matthew Cordes and David Hume. Stability and the M orse boundary. J. Lond. Math. Soc. (2) , 95(3):963--988, 2017

    2017

  9. [17]

    Morse boundaries of proper geodesic metric spaces

    Matthew Cordes. Morse boundaries of proper geodesic metric spaces. Groups Geom. Dyn. , 11(4):1281--1306, 2017

    2017

  10. [18]

    Small cancellation theory and burnside problem

    R\'emi Coulon. Small cancellation theory and burnside problem. Internat. J. Algebra Comput. , 24(3):251--345, 2014

    2014

  11. [19]

    Regularity of M orse geodesics and growth of stable subgroups

    Matthew Cordes, Jacob Russell, Davide Spriano, and Abdul Zalloum. Regularity of M orse geodesics and growth of stable subgroups. arXiv:2008.06379, 2020

    2008 arXiv

  12. [20]

    Contracting boundaries of CAT(0) spaces

    Ruth Charney and Harold Sultan. Contracting boundaries of CAT(0) spaces. J. Topol. , 8(1):93--117, 2015

    2015

  13. [21]

    Sous-groupes distingu\' e s et quotients des groupes hyperboliques

    Thomas Delzant. Sous-groupes distingu\' e s et quotients des groupes hyperboliques. Duke Math. J. , 83(3):661--682, 1996

    1996

  14. [22]

    A combination theorem for A nosov subgroups

    Subhadip Dey, Michael Kapovich, and Bernhard Leeb. A combination theorem for A nosov subgroups. Math. Z. , 293(1-2):551--578, 2019

    2019

  15. [23]

    Divergence in lattices in semisimple L ie groups and graphs of groups

    Cornelia Dru t u, Shahar Mozes, and Mark Sapir. Divergence in lattices in semisimple L ie groups and graphs of groups. Trans. Amer. Math. Soc. , 362(5):2451--2505, 2010

    2010

  16. [24]

    Corrigendum to `` D ivergence in lattices in semisimple L ie groups and graphs of groups'' [ MR 2584607]

    Cornelia Dru t u, Shahar Mozes, and Mark Sapir. Corrigendum to `` D ivergence in lattices in semisimple L ie groups and graphs of groups'' [ MR 2584607]. Trans. Amer. Math. Soc. , 370(1):749--754, 2018

    2018

  17. [25]

    Tree-graded spaces and asymptotic cones of groups

    Cornelia Dru t u and Mark Sapir. Tree-graded spaces and asymptotic cones of groups. Topology , 44(5):959--1058, 2005. With an appendix by Denis Osin and Mark Sapir

    2005

  18. [26]

    Durham and Samuel J

    Matthew G. Durham and Samuel J. Taylor. Convex cocompactness and stability in mapping class groups. Algebr. Geom. Topol. , 15(5):2839--2859, 2015

    2015

  19. [27]

    Spencer Dowdall and Samuel J. Taylor. Hyperbolic extensions of free groups. Geom. Topol. , 22(1):517--570, 2018

    2018

  20. [28]

    Regular languages for contracting geodesics

    Josh Eike and Abdalrazzaq Zalloum. Regular languages for contracting geodesics. arXiv:1809.02692, 2018

    2018 arXiv

  21. [29]

    Morse geodesics in torsion groups

    Elisabeth Fink. Morse geodesics in torsion groups. arXiv:1710.11191

    arXiv

  22. [30]

    Convex cocompact subgroups of mapping class groups

    Benson Farb and Lee Mosher. Convex cocompact subgroups of mapping class groups. Geom. Topol. , 6:91--152, 2002

    2002

  23. [31]

    Ping-pong on negatively curved groups

    Rita Gitik. Ping-pong on negatively curved groups. J. Algebra , 217(1):65--72, 1999

    1999

  24. [32]

    Hyperbolic groups

    Mikhail Gromov. Hyperbolic groups. In Essays in group theory , volume 8 of Math. Sci. Res. Inst. Publ. , pages 75--263. Springer, New York, 1987

    1987

  25. [33]

    Word hyperbolic extensions of surface groups

    Ursula Hamenst\" a dt. Word hyperbolic extensions of surface groups. arXiv:math/0505244

    arXiv

  26. [34]

    Stability in outer space

    Ursula Hamenst\" a dt and Sebastian Hensel. Stability in outer space. Groups Geom. Dyn. , 12(1):359--398, 2018

    2018

  27. [35]

    Christopher Hruska

    G. Christopher Hruska. Relative hyperbolicity and relative quasiconvexity for countable groups. Algebr. Geom. Topol. , 10(3):1807--1856, 2010

    2010

  28. [36]

    Nikolai V. Ivanov. Subgroups of T eichm\" u ller modular groups , volume 115 of Translations of Mathematical Monographs . American Mathematical Society, Providence, RI, 1992. Translated from the Russian by E. J. F. Primrose and revised by the author

    1992

  29. [37]

    Kent and Christopher J

    Autumn E. Kent and Christopher J. Leininger. Shadows of mapping class groups: C apturing convex cocompactness. Geom. Funct. Anal. , 18(4):1270--1325, 2008

    2008

  30. [38]

    Virtual amalgamation of relatively quasiconvex subgroups

    Eduardo Mart\' nez-Pedroza and Alessandro Sisto. Virtual amalgamation of relatively quasiconvex subgroups. Algebr. Geom. Topol. , 12(4):1993--2002, 2012

    1993

  31. [39]

    Mousley and Jacob Russell

    Sarah C. Mousley and Jacob Russell. Hierarchically hyperbolic groups are determined by their M orse boundaries. Geometriae Dedicata , Nov 2018

    2018

  32. [40]

    Ol shanskii, Denis V

    Alexander Yu. Ol shanskii, Denis V. Osin, and Mark V. Sapir. Lacunary hyperbolic groups. Geom. Topol. , 13(4):2051--2140, 2009. With an appendix by Michael Kapovich and Bruce Kleiner

    2009

  33. [41]

    An algorithm detecting hyperbolicity

    Panos Papasoglu. An algorithm detecting hyperbolicity. In Geometric and computational perspectives on infinite groups ( M inneapolis, MN and N ew B runswick, NJ , 1994) , volume 25 of DIMACS Ser. Discrete Math. Theoret. Comput. Sci. , pages 193--200. Amer. Math. Soc., Providen...

    1994

  34. [42]

    Jacob Russell, Davide Spriano, and Hung C. Tran. Convexity in hierarchcially hyperbolic spaces. arXiv:1809.09303, 2018

    2018 arXiv

  35. [43]

    On metric relative hyperbolicity

    Alessandro Sisto. On metric relative hyperbolicity. Preprint. arXiv:1210.8081

    arXiv

  36. [44]

    Projections and relative hyperbolicity

    Alessandro Sisto. Projections and relative hyperbolicity. Enseign. Math. (2) , 59(1-2):165--181, 2013

    2013

  37. [45]

    Hung C. Tran. On strongly quasiconvex subgroups. Geom. Topol. , 23(3):1173--1235, 2019

    2019

  38. [46]

    One-ended subgroups of graphs of free groups with cyclic edge groups

    Henry Wilton. One-ended subgroups of graphs of free groups with cyclic edge groups. Geom. Topol. , 16(2):665--683, 2012

    2012

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