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Algorithms detecting stability and Morseness for finitely generated groups

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Pith's one-line read Stability and Morseness of finitely generated subgroups are algorithmically detectable in mapping class groups, right-angled Artin groups, toral relatively hyperbolic groups, and limit-type groups.

desk verdict Real algorithmic contributions in the RAAG and relatively hyperbolic sections, but the mapping class group stability algorithm rests on a false characterization and needs repair. read the letter →

arxiv 1908.04460 v1 pith:5DZCCZDO submitted 2019-08-13 math.GR math.GT

Heejoung Kim This is my paper · ORCID
classification math.GRmath.GT MSC 20F6520F6720E0757M07
keywords stablesubgroupMorsemappingclassgroupright-angledArtinlimitcurvegraphrelativelyhyperbolicalgorithmictheory
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

This paper aims to turn stability and Morseness of finitely generated subgroups—two generalizations of quasiconvexity from word-hyperbolic groups—into properties that algorithms can certify for four families of groups. For mapping class groups it claims a partial algorithm that halts exactly on stable subgroups, a complete stability decision for undistorted subgroups, and a partial Morseness detector. For right-angled Artin groups it claims a complete stability decision and a partial Morseness detector, while for toral relatively hyperbolic groups it claims partial detectors plus a complete Morseness decision for undistorted subgroups. The same results specialize to finitely generated groups discriminated by a locally quasiconvex torsion-free hyperbolic group, including ordinary limit groups, where the algorithms are complete because every finitely generated subgroup is undistorted. A sympathetic reader would care because these are algorithmic results of a kind previously known only for quasiconvexity in word-hyperbolic groups.

What carries the argument

The load-bearing object is the orbit map into a δ-hyperbolic space with an algorithmic distance oracle: the curve graph for mapping class groups (vertices are isotopy classes of essential simple closed curves, edges record disjoint realizations), and the extension graph for right-angled Artin groups (vertices are conjugates of standard generators, edges record commutation), with the star metric providing a computable quasi-isometric model. The local-to-global principle for quasigeodesics in δ-hyperbolic spaces is the mechanism that lets an algorithm certify a global quasigeodesic by checking only paths of length at most a computable constant. For toral relatively hyperbolic groups the mechanism shifts to the induced peripheral structure of a relatively quasiconvex subgroup and its intersections with conjugates of peripheral subgroups; benign Dehn fillings are used to certify failures of peripheral finite index.

What would settle it

Run the first mapping-class-group stability algorithm on the trivial subgroup, using a base curve fixed by a nontrivial mapping class element: the trivial subgroup is stable, so a correct partial algorithm must eventually halt, whereas the proof's stated equivalence between stability and quasigeodesic images of every full-Cayley geodesic would predict failure on a long geodesic ending at that element; the observed behavior tests the termination claim.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is that stability of a finitely generated subgroup can be recognized by a uniform local-to-global check in a computable hyperbolic test space. In the mapping class group, the test space is the curve graph: the subgroup is stable exactly when its orbit is quasi-isometrically embedded, and the algorithm enumerates candidate quasigeodesic constants, checks all geodesic segments of length up to a computable bound starting at the identity, and terminates when the local checks pass. The same template works in right-angled Artin groups through the extension graph and the star metric. In toral relatively hyperbolic groups, stability and Morseness are instead characterized by intersections with peripheral subgroups, and the paper combines partial algorithms for computing those intersections with a Dehn-filling search that certifies failure of Morseness. Corollary E then yields complete algorithms for stability and Morseness in limit-type groups because every finitely generated subgroup there is undistorted.

Load-bearing premise

The algorithms assume that stability of a subgroup is exactly the geometric or peripheral condition they check—quasi-isometrically embedded orbit in a curve or extension graph, purely loxodromic behavior, or trivial intersections with peripheral subgroups—and that this characterization is effective enough that finitely many geodesic checks or intersection computations can certify it.

Editorial extensions

If this is right

  • In a mapping class group, every undistorted subgroup can be certified either stable or non-stable in finite time; the only gap for a complete Morseness decision is an algorithm for detecting infinite index.
  • In a right-angled Artin group, stability of any finitely generated subgroup is decidable in full; Morseness has a partial algorithm that halts on every Morse subgroup and runs forever only on non-Morse infinite-index subgroups.
  • In a toral relatively hyperbolic group, an undistorted subgroup can be certified Morse or non-Morse, and as a corollary the finite-index property for undistorted subgroups is decidable.
  • For ordinary limit groups, stability and Morseness of every finitely generated subgroup are completely decidable without an undistortedness hypothesis.
  • When these algorithms halt they also produce certificates: a quasiconvexity constant, a non-loxodromic witness, a nontrivial peripheral intersection, or a benign Dehn filling.

Reading between the lines

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

  • The local-to-global template is not tied to the four classes in the paper; any group with a computable hyperbolic test space and a stability-orbit characterization could in principle receive the same partial algorithm, though the paper does not claim this.
  • For the mapping class group proof, the algorithm's dependence on enumerating geodesics from the identity suggests that a practical implementation would first need an efficient curve-graph distance oracle and explicit hyperbolicity constants.
  • The right-angled Artin cube-complex algorithm connects stability to the absence of simple loops labeled by join words, pointing toward an automata-theoretic or regular-language description of stable subgroups that could extend to other subgroup properties.
  • If a complete algorithm for detecting infinite index in mapping class groups or right-angled Artin groups were found, the partial Morseness algorithms would become complete; the paper explicitly leaves this as an open problem.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 3 minor

Summary. The manuscript proposes algorithms for detecting stable and Morse subgroups of finitely generated groups in several classes. Theorem A gives a partial algorithm for stability of subgroups of mapping class groups, a complete algorithm for stability of undistorted subgroups, and a partial algorithm for Morseness. Theorem B gives a complete algorithm for stability in right-angled Artin groups and a partial algorithm for Morseness. Theorem C gives partial and complete algorithms for stability and Morseness in toral relatively hyperbolic groups, with Corollary D deciding finite index for undistorted subgroups and Corollary E applying to groups discriminated by locally quasiconvex torsion-free hyperbolic groups. The proofs reduce the target properties to previously known characterizations—convex cocompactness in the curve graph, pure loxodromicity in right-angled Artin groups, and Tran's intersection characterizations in relatively hyperbolic groups—and then invoke existing algorithmic tools.

Significance. If the gaps identified below are repaired, the paper would provide the first algorithmic recognition results for stable and Morse subgroups in mapping class groups and toral relatively hyperbolic groups, and a complete recognition algorithm for stability in right-angled Artin groups. The use of prior characterizations is systematic, and the Dehn-filling strategy for Theorem C(iv) is nontrivial and plausible. The cube-complex algorithm for Theorem B(i) in Section 3.2 appears to be a genuine algorithmic contribution. However, the current proof of Theorem A(i) rests on a false equivalence, and the first algorithm for Theorem B(i) has the same defect; since Theorem A is one of the paper's headline results, the manuscript requires substantial revision.

major comments (2)
  1. [Section 2, first proof of Theorem A(i)] The equivalence asserted in the fourth paragraph of this proof—that H is stable in Mod(S) iff there exist λ≥1, ε≥0 such that for every geodesic w in ΓMod(S) the image f(w) is a (λ,ε)-quasigeodesic in C(S)—is false. Theorem 2.2(2) only guarantees that the H-orbit is quasi-isometrically embedded in C(S); it does not constrain geodesics of Mod(S) outside H. To see the failure, fix a finite generating set B of Mod(S) that contains a Dehn twist τ with τ·x=x. For the stable subgroup H={1}, a geodesic representative of τ^n has length L_n→∞, and its image under f is a path in C(S) from x to x of length at least L_n, so it cannot be a (λ,ε)-quasigeodesic for fixed λ,ε as n→∞. The algorithm therefore checks a condition that fails for a stable subgroup and will never terminate on H={1}. This invalidates the proofs of Theorem A(i), A(ii), and A(iii) for the full class of surfaces; the second proof in Section 2.2 covers only closed hyperbolic surfaces. The proof should apply the local-to-global check to geodesics in the Cayley graph of H with the given generating set, not to all geodesics of Mod(S). There is also a notational ambiguity: A is introduced as a generating set for H, but ΓMod(S) denotes the Cayley graph of Mod(S); on the literal reading, one must extend A to a generating set of Mod(S), and with that reading the displayed equivalence is false.
  2. [Section 3.1, first proof of Theorem B(i)] The same type of false global condition appears here. The proof asserts that H is stable in AΓ iff there exist λ,ε such that every geodesic w in (ΓAΓ,d) is mapped to a (λ,ε)-quasigeodesic in Γ^e, and then reduces this to the condition that every geodesic starting from 1 in (ΓAΓ,d) is a quasigeodesic in (ΓAΓ,d_*). But Theorem 3.6 concerns only the H-orbit in Γ^e, and the identity map from the word metric d to the star metric d_* is not a quasi-isometry. For the connected anti-connected path graph a-b-c, the word w=a^n c a^{-n} is a geodesic in AΓ of length 2n+1, while its star length is 1 because the entire word lies in St(b). Since the trivial subgroup is stable, the claimed condition fails for a stable subgroup, so this first algorithm for Theorem B(i) is invalid. The second algorithm in Section 3.2 appears to be a valid complete algorithm for Theorem B(i), so this error does not by itself overturn Theorem B, but the first algorithm must be corrected or removed.
minor comments (3)
  1. [Theorem 2.3] Theorem 2.3 is missing the predicate: it should read 'A finitely generated subgroup H of Mod(S) is Morse if and only if either H is stable in Mod(S) or H has finite index in Mod(S)'.
  2. [Section 4.2, proof of Theorem C(iv)] In the benign-filling criterion, the indices are mixed: since the tuple contains γ∈H^{g_j}∩P_j, the intersection to check should be π(H)^{π(g_j)}∩π(P_j) and the quotient that must be infinite should be π(P_j)=P_j/N_j; the printed 'π(P_i)' appears to be a typo.
  3. [Throughout] There are numerous typographical errors, including 'qausigeodesics' (Definition 1.3), 'undisto rted' (Introduction), 'hyerbolic' (Section 4), 'patrial' (Questions 1.7 and 1.8), 'Moreseness' (Question 1.8), 'Cayely graph' (Section 3.1), and an extra closing bracket in the displayed quotient in Theorem 4.22. A thorough copyedit is needed.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the algorithmic reductions rely on external published characterizations, and the sole self-citation ([K19]) is prior independent support rather than an input-output loop.

full rationale

The paper's derivation chain is a sequence of reductions to previously proven characterizations and algorithmic subroutines: Theorem 2.2 (KL08/H05, BBKL18, DT15, K19/RST18), Theorem 3.6 (KMT17), Theorem 3.7 (T17, G17), Theorem 4.4 and Corollary 4.5 (T17), Theorem 4.13/4.14 (KMW17), Proposition 4.21-4.28 (GM17/O07), and Kapovich's quasiconvexity algorithm (K96). None of these results is defined in terms of the target algorithms, and none is fitted to the present paper's conclusions. The only self-citation, [K19], supplies the parameter-free characterization 'Morse iff stable or finite index' used in Theorem A(iii); it is a published result with stated assumptions that do not include the algorithmic conclusion, so under the hard rules it counts as independent support and does not raise the circularity score. The first proof of Theorem A(i) contains a serious mathematical gap—the asserted equivalence with quasigeodesicity of f(w) for every geodesic w in the full Cayley graph fails even for the trivial subgroup when a Dehn twist appears among the generators—but that is a correctness defect, not a circularity defect: the derivation does not covertly assume its own conclusion. Thus no circular pattern from the enumerated list is present.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No fitted constants appear; the paper is purely mathematical. The central claims rest on known characterizations and algorithmic results cited from the literature, plus one extension claim (Theorem 4.13 to abelian peripherals) that is not proved.

assumptions (6)
  • standard math Stability in Mod(S) is equivalent to convex cocompactness, to quasi-isometric embedding of the orbit into the curve graph, to being undistorted and purely pseudo-Anosov, and to being Morse of infinite index (Theorem 2.2).
    Cited to Kent-Leininger, Hamenstadt, Bestvina-Bromberg-Kent-Leininger, Durham-Taylor, and the author's own [K19], [RST18]. Used to reduce Theorem A to checking orbits in the curve graph.
  • standard math A subgroup H of A_Γ is stable if and only if it is purely loxodromic, equivalently any orbit map into the extension graph is a quasi-isometric embedding (Theorem 3.6, Koberda-Mangahas-Taylor).
    Primary characterization behind both proofs of Theorem B(i).
  • standard math The star metric on A_Γ is quasi-isometric to the extension graph Γ^e (Theorem 3.9, Kim-Koberda).
    Used in the first proof of Theorem B(i) to replace the extension graph by the star metric. The reviewer notes that the overstrong geodesic check in that proof is not justified by this quasi-isometry.
  • standard math For a toral relatively hyperbolic group, an undistorted subgroup is relatively quasiconvex, and stability and Morseness are characterized by the intersections H ∩ P^g being trivial or finite index (Theorem 4.4, Corollaries 4.5 and 4.7, Tran and Hruska).
    Basis for all of Section 4.
  • domain assumption The partial algorithms of Kharlampovich-Myasnikov-Weil exist for detecting relative quasiconvexity with peripherally finite index and computing intersections of such subgroups, and extend from toral relatively hyperbolic groups to relatively hyperbolic groups with finitely generated abelian…
    The extension beyond toral groups is asserted without proof; this is an unflagged assumption.
  • standard math Groves-Manning results on relatively hyperbolic Dehn fillings: sufficiency of long and H-wide fillings, injectivity of induced fillings, and peripheral structure preservation (Propositions 4.21, 4.24, 4.25).
    Used in the construction proving Proposition 4.28 and the algorithm in Theorem C(iv).

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Pith. "Pith review of Algorithms detecting stability and Morseness for finitely generated groups." pith.science (2026). https://pith.science/paper/5DZCCZDO

@misc{pith2026190804460,
  author       = {Pith},
  title        = {Pith review of: Algorithms detecting stability and Morseness for finitely generated groups},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5DZCCZDO}},
  note         = {Machine review of arXiv:1908.04460}
}
abstract

The notions of stable and Morse subgroups of finitely generated groups generalize the concept of a quasiconvex subgroup of a word-hyperbolic group. For a word-hyperbolic group $G$, Kapovich provided a partial algorithm which, on input a finite set $S$ of $G$, halts if $S$ generates a quasiconvex subgroup of $G$ and runs forever otherwise. In this paper, we give various detection and decidability algorithms for stability and Morseness of a finitely generated subgroup of mapping class groups, right-angled Artin groups, toral relatively hyperbolic groups, and finitely generated groups discriminated by a locally quasiconvex torsion-free hyperbolic group (for example, ordinary limit groups).

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Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. On the generalized membership problem in relatively hyperbolic groups

    math.GR 2019-08 conditional novelty 3.0 of 10

    For finitely presented relatively hyperbolic groups with well-behaved peripheral subgroups, the generalized membership problem is decidable for relatively quasi-convex subgroups.

Reference graph

Works this paper leans on

56 extracted references · 54 canonical work pages · cited by 1 Pith paper

  1. [1]

    Aougab, Uniform Hyperbolicity of the Graphs of Curves, Geom

    T. Aougab, Uniform Hyperbolicity of the Graphs of Curves, Geom. Topol. 17 (2013) 2855–-2875

    work page 2013

  2. [2]

    Bestvina, K

    M. Bestvina, K. Bromberg, R. Kent, and C. Leininger, Undistorted purely pseudo-Anosov group, J. Reine Agnew Math. doi:10.1515/crelle-2018-0013 (2018)

    doi:10.1515/crelle-2018-0013 2018

  3. [3]

    Behrstock and R

    J. Behrstock and R. Charney, Divergence and quasimorphisms of right-angled Artin groups, Math. Ann. 352 (2) (2012) 339–-356

    work page 2012

  4. [4]

    Bestvina and M

    M. Bestvina and M. Handel, Train-tracks for surface homeomorphisms, Topology. 34(1) (1995) 109--140

    work page 1995

  5. [5]

    Bridson and A

    M. Bridson and A. Haefliger, Metric spaces of non-positive curvature, Grundlehren der Mathematischen Wissenschaften, 319. Springer-Verlag, Berlin, 1999

    work page 1999

  6. [6]

    Bigdely and D

    H. Bigdely and D. Wise, Quasiconvexity and Relatively Hyperbolic Groups That Split, Michigan Math. J. 62 (2013) 387–-406

    work page 2013

  7. [7]

    Birman, D

    J. Birman, D. Margalit, and W. Menasco, Efficient geodesics and an effective algorithm for distance in the complex of curves, Math. Ann. 366(3-4) (2016) 1253–-1279

    work page 2016

  8. [8]

    Bowditch, Relatively hyperbolic groups, Preprint, Univ

    B. Bowditch, Relatively hyperbolic groups, Preprint, Univ. of Southampton, 1999

    work page 1999

Show all 56 references
  1. [9]

    Bowditch, Intersection numbers and the hyperbolicity of the curve complex, J

    B. Bowditch, Intersection numbers and the hyperbolicity of the curve complex, J. Reine Angew. Math. 598 (2006) 105–-129

    2006

  2. [10]

    Bowditch, Uniform hyperbolicity of the curve graphs, (2012) available at http://homepages.warwick.ac.uk/ masgak/papers/uniformhyp.pdf

    B. Bowditch, Uniform hyperbolicity of the curve graphs, (2012) available at http://homepages.warwick.ac.uk/ masgak/papers/uniformhyp.pdf

    2012

  3. [11]

    Bridson and D

    M. Bridson and D. Wise, Malnormality is undecidable in hyperbolic groups, Israel J. Math. 124 (2001) 313–-316

    2001

  4. [12]

    Charney, An introduction to right-angled Artin groups, Geom

    R. Charney, An introduction to right-angled Artin groups, Geom. Dedicata. 125 (2007) 141--158

    2007

  5. [13]

    M. Clay, K. Rafi, and S. Schleimer, Uniform hyperbolicity of the curve graph via surgery sequences, Algebr. Geom. Topol. 14 (2014) 3325–-3344

    2014

  6. [14]

    Dahmani, Combination of convergence groups, Geom

    F. Dahmani, Combination of convergence groups, Geom. Topol. 7 (2003) 933-–963

    2003

  7. [15]

    Diekert, J

    V. Diekert, J. Kausch, and M. Lohrey, Logspace computations in Coxeter groups and graph groups, Computational and combinatorial group theory and cryptography, 77-–94, Contemp. Math., 582, Amer. Math. Soc., Providence, RI, 2012

    2012

  8. [16]

    Drutu and M

    C. Drutu and M. Sapir, Tree-graded spaces and asymptotic cones of groups, Topology. 44 (2005) 959–-1058

    2005

  9. [17]

    Durham and S

    M. Durham and S. Taylor, Convex cocompactness and stability in mapping class groups, Algebr. Geom. Topol. 15(5) (2015) 2839–-2859

    2015

  10. [18]

    Epstein, J

    D. Epstein, J. Cannon, D. Holt, S. Levy, M. Paterson, and W. Thurston, Word Processing in Groups, Jones and Bartlett, 1992

    1992

  11. [19]

    Farb and L

    B. Farb and L. Mosher, Convex cocompact subgroups of mapping class groups, Geom. Topol. 6 (2002) 91–-152

    2002

  12. [20]

    Genevois, Hyperbolicities in CAT(0) cube complexes, arXiv preprint arXiv:1709.08843

    A. Genevois, Hyperbolicities in CAT(0) cube complexes, arXiv preprint arXiv:1709.08843

    arXiv

  13. [21]

    Gersten and H

    S. Gersten and H. Short, Rational Subgroups of Biautomatic Groups, Ann. Math. 134 (1991) 125--158

    1991

  14. [22]

    Gromov, Hyperbolic Groups

    M. Gromov, Hyperbolic Groups. In: Essays in Group Theory, S.M. Gersten (ed.) MSRI series 8, Springer: Berlin Heidelberg New York, 75--263

  15. [23]

    Groves and J

    D. Groves and J. Manning, Dehn filling in relatively hyperbolic groups, Israel J. Math. 168 (2008) 317–-429

    2008

  16. [24]

    Groves and J

    D. Groves and J. Manning, Quasiconvexity and Dehn filling, arXiv preprint arXiv:1708.07968v3 (To appear in Amer. J. Math)

    arXiv

  17. [25]

    Haglund, Finite index subgroups of graph products, Geom

    F. Haglund, Finite index subgroups of graph products, Geom. Dedicata. 135(1) (2008) 167–-209

    2008

  18. [26]

    Hamenst a dt, Word hyperbolic extensions of surface groups, arXiv preprint arXiv:0505244

    U. Hamenst a dt, Word hyperbolic extensions of surface groups, arXiv preprint arXiv:0505244

  19. [27]

    Hamenst a dt, Geometry of the complex of curves and of Teichm u ller space, Handbook of Teichm\" u ller theory, 1 EMS (2007) 447–-467

    U. Hamenst a dt, Geometry of the complex of curves and of Teichm u ller space, Handbook of Teichm\" u ller theory, 1 EMS (2007) 447–-467

    2007

  20. [28]

    Hensel, P

    S. Hensel, P. Przytycki, and R. Webb, 1-slim triangles and uniform hyperbolicity for arc graphs and curve graphs, J. Eur. Math. Soc. (JEMS) 17(4) (2015) 755-–762

    2015

  21. [29]

    Hermiller and J

    S. Hermiller and J. Meier, Algorithms and geometry for graph products of groups, J. Algebra. 171(1) (1995) 230–-257

    1995

  22. [30]

    Hruska, Relative hyperbolicity and relative quasiconvexity for countable groups, Algebr

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

    2010

  23. [31]

    Kapovich, Detecting Quasiconvexity: Algorithmic Aspects, Geometric and Computational Perspectives on Infinite Groups, DIMACS Ser

    I. Kapovich, Detecting Quasiconvexity: Algorithmic Aspects, Geometric and Computational Perspectives on Infinite Groups, DIMACS Ser. Discrete Math. Theoret. Comput. Sci. 25 (1996) 91–-99

    1996

  24. [32]

    Kapovich and A

    I. Kapovich and A. Myasnikov, Stallings foldings and the subgroup structure of free groups, J. Algebra. 248(2) (2002) 608–-668

    2002

  25. [33]

    Kapovich, R

    I. Kapovich, R. Weidmann, and A. Miasnikov, Foldings, graphs of groups and the membership problem, Internat. J. Algebra Comput. 15(1) (2005) 95–-128

    2005

  26. [34]

    Kent and C

    A. Kent and C. Leininger, Shadows of mapping class groups: capturing convex cocompactness, Geom. Funct. Anal. 18(4) (2008) 1270–-1325

    2008

  27. [35]

    Kharlampovich and A

    O. Kharlampovich and A. Myasnikov, Limits of relatively hyperbolic groups and Lyndon's completions (English summary), J. Eur. Math. Soc. (JEMS) 14(3) (2012) 659–-680

    2012

  28. [36]

    Kharlampovich, A

    O. Kharlampovich, A. Miasnikov, and P. Weil, Stallings Graphs for Quasi-Convex Subgroups, J. Algebra. 488 (2017) 442--483

    2017

  29. [37]

    Kharlampovich and P

    O. Kharlampovich and P. Weil, On the generalized membership problem in relatively hyperbolic groups, arXiv preprint arXiv:1908.03525

    1908 arXiv

  30. [38]

    Kim, Stable subgroups and Morse subgroups in mapping class groups, Internat

    H. Kim, Stable subgroups and Morse subgroups in mapping class groups, Internat. J. Algebra Comput. 29(5) (2019) 893--903

    2019

  31. [39]

    Kim and T

    S. Kim and T. Koberda, Embedability between right-angled Artin groups, Geom. Topol. 17(1) (2013) 493–-530

    2013

  32. [40]

    Kim and T

    S. Kim and T. Koberda, The geometry of the curve graph of a right-angled Artin group, Internat. J. Algebra Comput. 24 (2014) 121–-169

    2014

  33. [41]

    Koberda, J

    T. Koberda, J. Mangahas and S. Taylor, The geometry of purely loxodromic subgroups of right-angled Artin groups, Trans. Amer. Math. Soc. 369 (2017) 8179–-8208

    2017

  34. [42]

    Leasure, Geodesics in the complex of curves of a surface, The University of Texas at Austin, ProQuest Dissertations Publishing, 2002

    J. Leasure, Geodesics in the complex of curves of a surface, The University of Texas at Austin, ProQuest Dissertations Publishing, 2002. 3114766

    2002

  35. [43]

    Lyndon and P

    R. Lyndon and P. Schupp, Combinatorial Group Theory, 1977, Springer, Berlin, Hei- delberg, New York

    1977

  36. [44]

    Mangahas and S

    J. Mangahas and S. Taylor, Convex cocompactness in mapping class groups via quasiconvexity in right-angled Artin groups, Proc. London Math. Soc. 12(5) (2016) 855–-881

    2016

  37. [45]

    Manning and E

    J. Manning and E. Martinez-Pedroza, Separation of relatively quasi-convex subgroups, Pacific J. Math. 244(2) (2010) 309–-334

    2010

  38. [46]

    Mosher, Mapping class groups are automatic, Ann

    L. Mosher, Mapping class groups are automatic, Ann. of Math. 142 (1995) 303-384

    1995

  39. [47]

    Osin, Relatively hyperbolic groups: intrinsic geometry, algebraic properties, and algorithmic problems, Mem

    D. Osin, Relatively hyperbolic groups: intrinsic geometry, algebraic properties, and algorithmic problems, Mem. Amer. Math. Soc. 179(843) (2006) 1--100

    2006

  40. [48]

    Osin, Peripheral fillings of relatively hyperbolic groups, Invent

    D. Osin, Peripheral fillings of relatively hyperbolic groups, Invent. Math. 167(2) (2007) 295–-326

    2007

  41. [49]

    Papasoglu, Strongly geodesically automatic groups are hyperbolic, Invent

    P. Papasoglu, Strongly geodesically automatic groups are hyperbolic, Invent. Math. 121(2) (1995) 323–-334

    1995

  42. [50]

    Lyndon and P

    R. Lyndon and P. Schupp, Combinatorial group theory. Classics in Mathematics, Springer-Verlag, Berlin, 2001. Reprint of the 1977 edition

    2001

  43. [51]

    Russell, D

    J. Russell, D. Spriano, and H. Tran, Convexity in Hierarchically Hyperbolic Spaces, arXiv preprint arXiv:1809.09303

    arXiv

  44. [52]

    Servatius, Automorphisms of graph groups, J

    H. Servatius, Automorphisms of graph groups, J. Algebra. 126(1) (1989) 34–-60

    1989

  45. [53]

    Shackleton, Tightness and computing distances in the curve complex, Geom

    K. Shackleton, Tightness and computing distances in the curve complex, Geom. Dedicata. 160 (2012) 243–-259

    2012

  46. [54]

    Tran, On strongly quasiconvex subgroups, Geom

    H. Tran, On strongly quasiconvex subgroups, Geom. Topol. 23(3) (2019) 1173–-1235

    2019

  47. [55]

    Watanabe, Intersection numbers in the curve graph with a uniform constant, J

    Y. Watanabe, Intersection numbers in the curve graph with a uniform constant, J. Topol. Anal. 9(3) (2017) 419-–439

    2017

  48. [56]

    Webb, Combinatorics of tight geodesics and stable lengths, Trans

    R. Webb, Combinatorics of tight geodesics and stable lengths, Trans. Amer. Math. Soc. 367(10) (2015) 7323–-7342

    2015

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