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Research

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* My arXiv papers

Geometric Group Theory mailing list.

I currently administer a geometric group theory mailing list [email protected] for announcements about events and positions related to geometric group theory. To submit such an announcement, e-mail it directly to [email protected]. The list’s homepage is located at this link. You can subscribe/unsubscribe to the list there (you can also do that by sending me an e-mail request at [email protected]).

New York Group Theory Seminar.

I am a co-organizer of the New York Group Theory Seminar at the CUNY Graduate Center. In the Fall 2024 semester the seminar meets in a hybrid format, with some in person talks and some online Zoom talks. The in-person talks are on Fridays at 4:15pm in room 5417 at the CUNY Graduate Center. The onine Zoom talks are on Fridays at 4:00pm U.S. eastern time. I also maintain the NYGT Seminar mailing list. If you would like to receive the weekly announcements, including Zoom links and passwords, for the seminar talks, please e-mail me (from an academic institution affiliated e-mail address) a request to [email protected]. Videos of prior NYGT Seminar talks are available at the NYGT Youtube channel.

Publications.

I am currently supported by the NSF grant DMS-1905641

    1. Ilya Kapovich, On Dehn functions for infinite group presentations, arXiv:2606.01290
    2. Ilya Kapovich, On quantitative aspects of trace polynomials,  arXiv:2605.25265
    3. Ilya Kapovich, Alexei Myasnikov and Vladimir Shpilrain, Combinatorial Group Theory: Open Problems on Infinite Groups, De Gruyter, 2026, ISBN:9783112208137, 9783119146791. https://doi.org/10.1515/9783112208137
    4. Ilya Kapovich, “On the Hausdorff dimension and attracting laminations for fully irreducible automorphisms of free groups,” preprint, 2024; arXiv:2410.02058
    5. Naomi Andrew, Edgar A. Bering IV, Ilya Kapovich, Stefano Vidussi; with an appendix by Peter Shalen, “On two-generator subgroups of mapping torus groups,” Journal of the London Mathematical Society 112 (2025), no. 1, Paper No. e70226.
    6. “Counting conjugacy classes of fully irreducibles: double exponential growth”, (joint with Catherine Pfaff); Geometriae Dedicata 218 (2024), no. 2, Paper no. 39; DOI: 10.1007/s10711-024-00885-4
    7. “Ascending chain condition in generic groups,” preprint, 2023; arXiv:2306.06706
    8. “Wolfgang Haken, 1928-2022”, (joint with Marc Lackenby, Peter Shalen, Robin Wilson and Patrick Callahan), Notices of the American Mathematical Society, 70 (2023), no. 9, 1452-1467
    9. “Primitivity rank for random elements in free groups,” Journal of Group Theory 25 (2022), no. 5, 823-835
    10. “Primitivity index bounds in free groups, and the second Chebyshev function” (joint with Zachary Simon) International Journal of Algebra and Computation 33 (2023), no. 1, pp. 105-122
    11. “Complexity and Randomness in Group Theory. GAGTA BOOK 1.” (By Frederique Bassino, Ilya Kapovich, Markus Lohrey, Alexei Miasnikov, Cyril Nicaud, Andrey Nikolaev, Igor Rivin, Vladimir Shpilrain, Alexander Ushakov, Pascal Weil), De Gruyter, 2020; ISBN 9783110667028
    12. “Random trees in the boundary of Outer space” , (joint with Joseph Maher, Catherine Pfaff, and Samuel Taylor); Geometry & Topology 26 (2022), no. 1, 127-162
    13. “Generic-case complexity of Whitehead’s algorithm, revisited “, preprint arXiv:1903.07040
    14. “Random outer automorphisms of free groups: Attracting trees and their singularity structures”, (joint with Joseph Maher, Catherine Pfaff, and Samuel Taylor); Transactions of the American Mathematical Society 375 (2022), no. 1, pp. 525-557
    15. “Counting conjugacy classes in Out(FN)“, (joint with Michael Hull), Bulletin of the Australian Mathematical Society 97 (2018), no 3, 412-421
    16. “Stable Strata of Geodesics in Outer Space”, (joint with Yael Algom-Kfir and Catherine Pfaff), Internat. Math. Res. Notices (IMRN) 2019 (2019), no. 14, pp. 4549-4578
    17. “Detecting fully irreducible automorphisms: a polynomial time algorithm. With an appendix by Mark C. Bell”, Experimental Mathematics, 28 (2019), no. 1, pp. 24-38
    18. “Endomorphisms, train track maps, and fully irreducible monodromies “, (joint with Spencer Dowdall and Christopher Leininger), Groups, Geometry, and Dynamics 11 (2017), no. 4, pp. 1179-1200
    19. “The primitivity index function for a free group, and untangling closed curves on hyperbolic surfaces. With an appendix by Khalid Bou-Rabee.” (joint with Neha Gupta), Mathematical Proceedings of the Cambridge Philosophical Society 166 (2019), no. 1, pp. 83-121 Published version at the publisher’s website
    20. “McMullen polynomials and Lipschitz flows for free-by-cyclic groups” , (joint with Spencer Dowdall and Christopher J. Leininger); Journal of the European Mathematical Society (JEMS) 19 (2017), no. 11, pp. 3253-3353
    21. “An integral weight realization theorem for subset currents on free groups” , Topology Proceedings 50 (2017), pp. 213-236
    22. “Wolfgang Haken: A biographical sketch”, Illinois Journal of Mathematics 60 (2016), no. 1, pp. 3-9; special issue in honor of Wolfgang Haken
    23. “Cannon-Thurston maps for hyperbolic free group extensions”, (joint with Spencer Dowdall and Samuel Taylor), Israel J. Math. 216 (2016), no. 2, pp. 753-797
    24. “Musings on generic-case complexity”, preprint
    25. “On purely loxodromic actions” , Monatshefte fur Mathematik 181 (2016), no. 1, pp. 89-101
    26. “Conical limit points and the Cannon-Thurston map” (joint with Woojin Jeon, Christopher Leininger and Ken’ichi Ohshika), Conformal Geometry and Dynamics, 20 (2016), pp. 58-80
    27. “Nielsen equivalence in a class of random groups” (joint with Richard Weidmann), Journal of Topology 9 (2016), no. 2, 502-534
    29. “Patterson-Sullivan currents, generic stretching factors and the asymmetric Lipschitz metric for Outer space”, (joint with Martin Lustig), Pacific J. Math. 277 (2015), no. 2, pp. 371-398
    30. “Dynamics on free-by-cyclic groups” , (joint with Spencer Dowdall and Christopher J. Leininger); Geometry and Topology 19 (2015), 2801-2899
    31. “A train track directed random walk on $Out(F_r)$” (joint with Catherine Pfaff), International Journal of Algebra and Computation 25 (2015), no. 5, pp. 745-798
    32. “Cannon-Thurston fibers for iwip automorphisms of $F_N$”, (joint with Martin Lustig), Jour. London Math. Soc. 91 (2015), no. 1, 203-224
    33. “Unbounded asymmetry of stretch factors”, (joint with Spencer Dowdall and Christopher J. Leininger); Comptes Rendus Mathematique, 352 (2014), no. 11, 885-887
    34. “Algorithmic detectability of iwip automorphisms”, Bull. London Math. Soc. 46 (2014), no. 2, pp. 279-290; online version at the BLMS site available here
    35. “On hyperbolicity of free splitting and free factor complexes”, (joint with Kasra Rafi), Groups, Geometry and Dynamics, 8 (2014), no. 2, pp. 391-414
    36. “Quasi-isometric co-Hopficity of non-uniform lattices in rank-one semi-simple Lie groups”, (joint with Anton Lukyanenko), Conformal Geometry and Dynamics, 16 (2012), 269-282
    37. “Spectral rigidity of automorphic orbits in free groups”, (joint with Mathieu Carette, Stefano Francaviglia and Armando Martino), Algebraic and Geometric Topology 12 (2012), pp. 1457-1486
      “Corrigendum to ”Spectral rigidity of automorphic orbits in free groups””, (joint with Mathieu Carette, Stefano Francaviglia and Armando Martino);   Algebraic and Geometric Topology 14 (2014), no. 5, pp. 3081-3088
    38. “Subset currents on free groups”, (joint with Tatiana Nagnibeda), Geometriae Dedicata, 166 (2013), 307-348
    39. “Invariant laminations for irreducible automorphisms of free groups”, (joint with Martin Lustig), Quarterly J. Math. 65 (2014), no. 4, pp. 1241-1275  Q. J. Math. (2014) doi: 10.1093/qmath/hat056
    40. “Random length-spectrum rigidity for free groups”, Proceedings of AMS 140 (2012), no. 5, pp. 1549-1560
    41. “Stabilizers of R-trees with free isometric actions of FN “, (with Martin Lustig), Journal of Group Theory, 14 (2011), no. 5, pp. 673-694
    42. “Domains of proper discontinuity on the boundary of Outer space”,(with Martin Lustig), Illinois J. Math., 54 (2010), no. 1, pp. 89-108, special issue dedicated to Paul Schupp
    43. “On mathematical contributions of Paul E. Schupp”, Illinois J. Math., 54 (2010), no. 1, pp. 1-9, special issue dedicated to Paul Schupp
    44. “Ping-pong and Outer space”, (with Martin Lustig), Journal of Topology and Analysis 2 (2010), no. 2, pp. 173-201
    45. “Nielsen equivalence in small cancellation groups”, (with Richard Weidmann), preprint
    46. “Geometric entropy of geodesic currents on free groups”, (with Tatiana Nagnibeda), Dynamical Numbers: Interplay Between Dynamical Systems and Number Theory, Contemporary Mathematics series, American Mathematical Society, 2010, pp. 149-176
    47. “Intersection form, laminations and currents on free groups “, (with Martin Lustig), Geom. Funct. Anal. 19 (2010), no. 5, pp. 1426-1467
    48. “Geometric Intersection Number and analogues of the Curve Complex for free groups”, (with Martin Lustig), Geometry and Topology 13 (2009), pp. 1805-1833
    49. “Some Quantitative Aspects of Fractional Computability”, (with Paul Schupp), preprint
    50. “On group-theoretic models of randomness and genericity” (with Paul Schupp), Groups, Geometry and Dynamics 2 (2008), no. 3, pp. 383-404
    51. “The actions of $Out(F_k)$ on the boundary of Outer space and on the space of currents: minimal sets and equivariant incompatibility” (with Martin Lustig), Ergodic Theory and Dynamical Systems 27 (2007), no. 3, pp. 827-847
    52. “Random quotients of the modular group are rigid and essentially incompressible” (with Paul Schupp), Journal fur die reine und angewandte Mathematik (Crelle’s Journal), 628 (2009),  pp. 91-119
    53. “On the absence of McShane-type identities for the outer space” (with Igor Rivin), Journal of Algebra 320 (2008), no. 10, pp. 3659-3670
    54. “Clusters, currents and Whitehead’s algorithm”, Experimental Mathematics 16 (2007), no. 1, pp. 67-76
    55. ” Densities in free groups and $\mathbb{Z}^k$, Visible Points and Test Elements” (with Igor Rivin, Paul Schupp and Vladimir Shpilrain), Math. Research Letters 14 (2007), no. 2, pp. 263-284
    56. “The Subadditive Ergodic Theorem and generic stretching factors for free group automorphisms” (with Vadim Kaimanovich and Paul Schupp), Israel J. Math. 157 (2007), pp. 1-46
    57. “Translation equivalence in free groups”, (with Gilbert Levitt, Paul Schupp and Vladimir Shpilrain),  Transact. Amer. Math. Soc. 359 (2007), no. 4, pp. 1527-1546
    58. “The Patterson-Sullivan embedding and minimal volume entropy for outer space”, (joint with Tatiana Nagnibeda), Geom. Funct. Anal. (GAFA) 17 (2007), no 4, pp 1201-1236
    59. “Generic properties of Whitehead’s Algorithm and isomorphism rigidity of random one-relator groups” (with Paul Schupp and Vladimir Shpilrain), Pacific J. Math. 223 (2006), no. 1, pp. 113-140
    60. “Currents on free groups”,   Topological and Asymptotic Aspects of Group Theory (R. Grigorchuk, M. Mihalik, M. Sapir and Z. Sunik, Editors), AMS Contemporary Mathematics Series, vol. 394, 2006, pp. 149-176
    61. “The frequency space of a free group”, Internat. J. Alg. Comput. (special Gaeta Grigorchuk’s 50’s birthday issue), 15 (2005), no. 5-6, pp. 939-969
    62. “Delzant’s T-invariant, Kolmogorov complexity and one-relator groups” (with Paul Schupp), Commentari Math. Helv. 80 (2005), no. 4, pp. 911-933
    63. “Kleinian groups and the rank problem”, (with Richard Weidmann), Geometry and Topology 9 (2005), pp. 375-402
    64. “Foldings, graphs of groups and the membership problem” (with Richard Weidmann and Alexei Myasnikov), Internat. J. Alg. Comput. 15 (2005), no. 1, 95-128
    65. “Acylindrical accessibility for groups acting on R-trees” (with Richard Weidmann), Math. Zeitschrift 249 (2005), no. 4, pp. 773-782
    66. “Genericity, the Arzhantseva-Ol’shanskii method and the isomorphism problem for one-relator groups” (with Paul Schupp),  Mathematische Annalen 331(2005), no. 1, pp. 1-19
    67. “Average-case complexity and decision problems in group theory” (with Alexei Myasnikov, Paul Schupp and Vladimir Shpilrain), Advances in Mathematics 190 (2005), no. 2, pp. 343-359
    68. “Relative hyperbolicity and Artin groups”(with Paul Schupp),  Geom. Dedicata 107 (2004), no. 1, pp. 153-167
    69. “Freely indecomposable groups acting on hyperbolic spaces”(with Richard Weidmann),  Internat. J. Algebra and Comput. 14 (2004), no. 2, pp. 115-171
    70. “Bounded rank subgroups of Coxeter groups, Artin groups and one-relator groups with torsion” (with Paul Schupp), Proceedings of the London Mathematical Society 88 (2004), no. 1, pp. 89-113
    71. “Generic-case complexity, decision problems in group theory and random walks” (with Alexei Myasnikov, Paul Schupp and Vladimir Shpilrain),  J. Algebra 264 (2003), no. 2, pp. 665-694
    72. “Nielsen methods and groups acting on hyperbolic spaces” (with Richard Weidmann), Geometriae Dedicata 98 (2003), no. 1, pp. 95-121
    73. “The Frattini subgroup for subgroups of hyperbolic groups”,  J. Group Theory 6 (2003), no. 1, pp. 115-126
    74. “The non-amenability of Schreier graphs for infinite index quasiconvex subgroups of hyperbolic groups”, Ensign. Math. 48 (2002), no. 3-4, pp. 359-375
    75. “The geometry of relative Cayley graphs for subgroups of hyperbolic  groups”, preprint
    76. “A remark on mapping tori of free group endomorphisms”,  preprint
    77. “Boundaries of hyperbolic groups” (with Nadia Benakli), Combinatorial and Geometric Group Theory (R.Gilman, et al, editors), Contemporary Mathematics, vol. 296, 2002, pp. 39-94
    78. “A note on the Poenaru condition”,   J. Group Theory 5 (2002), no. 1,  pp. 119-127
    79. “Stallings foldings and the subgroup structure of free groups” (with Alexey Myasnikov), J. Algebra 248 (2002), no 2,   pp. 608-668, and an Addendum
    80. “The Combination Theorem and Quasiconvexity”,  Intern. J. Algebra Comput. 11 (2001), no. 2,  pp. 185-216
    81. “The subgroup properties of fully residually free groups” , Transact. Amer. Math. Soc. 354 (2002), no. 1,  pp. 335-362; and an ERRATUM ,  Transact. Amer. Math. Soc.  355 (2003), no. 3,  pp. 1295–1296
    82. “On the failure of the Co-Hopf property for subgroups of word-hyperbolic groups” (with Dani Wise),   Israel. J. Math. 122 (2001),  pp. 125-147
    83.  “Mapping tori of endomorphisms of free groups”,  Comm. in Algebra 28 (2000), no. 6,  pp. 2895-2917
    84. “Equivalence of some residual properties for word-hyperbolic groups”(with Dani Wise),   J. Algebra 223 (2000), no. 2, pp. 562-583
    85. ” Two-generated groups acting on Trees”(with Richard Weidmann), Archiv der Mathematik (Basel) 73 (1999), no. 3, pp. 172-181
    86. “On the structure of two-generated word-hyperbolic groups” (with Richard Weidmann),   Math. Z. 231 (1999), no. 4, pp. 783-80
    87. “One-relator groups and the Howson property”, Comm. in Algebra 27 (1999), no. 3,  pp. 1057-1072
    88. “A non-quasiconvexity embedding theorem for hyperbolic groups”,   Math Proc. Cambridge Phil. Soc. 127 (1999), no. 3,  pp. 461-486
    89. “Quasiconvexity and amalgams”, Intern. J. Alg. Computation 7 (1997), no. 6,  pp. 771-811
    90. “Amalgamated products and the Howson Property” , Canadian Math. Bulletin, 40 (1997), no. 3,  pp. 330-340
    91. “Small cancellation groups and translation numbers”, Trans. Amer. Math. Soc., 349 (1997), no. 5,  pp. 1851-1875
    92. “Greenberg’s theorem for quasiconvex subgroups of word hyperbolic groups” (with Hamish Short), Canadian J. Math., 48 (1996), no. 6, pp. 1224-1244
    93. “An example of a non-quasiconvex subgroup of a word hyperbolic group with an exotic limit set”,  New York J. Math., 1 (1995),  pp. 184-195
    94. “Detecting quasiconvexity: algorithmic aspects”,  AMS DIMACS series, 25,  Geometric and Computational Perspectives on Infinite Groups, 1995, pp. 91-100