My name is Ilya Kapovich,
and I am a Professor in the Department of Mathematics and Statistics at the Hunter College of CUNY, starting from with the Fall 2018 semester.

my photo

The above photo is licensed by under the Creative Commons Attribution-ShareAlike 4.0 license. The photo was taken by Ilya Kapovich in December 2012 at the University of Illinois in Urbana-Champaign.

Here is a link to my old webpage at the University of Illinois at Urbana-Champaign.

Here is a link to the webpage for the Spring 2017 Math 751 General Topology course I taught at Hunter.

My research papers in arXiv.

My March 15, 2017 talk at the Hunter College GRECS Seminar, ``Groups as geometric objects.

My office at Hunter is HE 917, tel. 212-772-5303

Here is a reasonably up-to-date version of my CV (April 14, 2019).

Starting from the Fall 2021 semester, I am the Math BA advisor (dealing with the Mathematics undergraduate majors) in the Department of Mathematics and Statistics at Hunter. If you have an advising question related to the Mathematics major or want to schedule an advising appointment, please e-mail me at During the Fall 2021 semester, because of the pandemic, all of my advising appointments will be conducted online, via Zoom.

In the Fall 2021 semester I am teaching Math 155,Calculus II, Section 05 (online), Tue, Fri, 9:10am-11am. Details about the course are available in Blackboard.

Contact information

e-mail [preferred way of contacting me]

office phone 212-772-5303

fax 212-772-4858

office location: Hunter East 917

Mailing address:

Department of Mathematics and Statistics
Room 919/944 East
Hunter College
695 Park Avenue
New York, NY 10065, U.S.A.

Geometric Group Theory mailing list.

I administer a geometric group theory mailing list for announcements about events and positions related to geometric group theory. To submit such an announcement, e-mail it directly to 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

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 2021 semester the seminar meets remotely, via Zoom, on Fridays at 4pm 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 Videos of prior NYGT Seminar talks are available at the NYGT Youtube channel.

Human rights work

*I served as the Chair of the AMS Committee on the Human Rights of Mathematicians (02/01/2020-01/31/2021)
If you have a matter requiring the committee's attention, please e-mail

*I am a member of the Steering Committee of the AAAS Science and Human Rights Coalition.

*AMS and ICM 2022, letter to the editor, Notices of the American Mathematical Society, volume 68, number 5 (May 2021), pp. 681-682

* The case of Azat Miftakhov

* Putin is distracting the world from his atrocities letter to the editor, Washington Post, February 21, 2022


Link to my papers posted at arxiv.og 

I am currently supported by the NSF grant DMS-1905641
  1. "Primitivity rank for random elements in free groups," Journal of Group Theory, to appear
  2. "Primitivity index bounds in free groups, and the second Chebyshev function" (joint with Zachary Simon); arXiv:2012.13655
  3. "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
  4. "Random trees in the boundary of Outer space" , (joint with Joseph Maher, Catherine Pfaff, and Samuel Taylor); arXiv:1904.10026; Geometry & Topology, to appear
  5. "Generic-case complexity of Whitehead's algorithm, revisited ", preprint arXiv:1903.07040
  6. "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
  7. "Counting conjugacy classes of fully irreducibles: double exponential growth", (joint with Catherine Pfaff); preprint arXiv:1801.07471
  8. "Counting conjugacy classes in Out(FN)", (joint with Michael Hull), Bulletin of the Australian Mathematical Society 97 (2018), no 3, 412-421
  9. "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
  10. "Detecting fully irreducible automorphisms: a polynomial time algorithm. With an appendix by Mark C. Bell", Experimental Mathematics, 28 (2019), no. 1, pp. 24-38
  11. "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
  12. "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
  13. "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
  14. "An integral weight realization theorem for subset currents on free groups" , Topology Proceedings 50 (2017), pp. 213-236
  15. "Wolfgang Haken: A biographical sketch", Illinois Journal of Mathematics 60 (2016), no. 1, pp. 3-9; special issue in honor of Wolfgang Haken
  16. "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
  17. "Musings on generic-case complexity", preprint
  18. "On purely loxodromic actions" , Monatshefte fur Mathematik 181 (2016), no. 1, pp. 89-101
  19. "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
  20. "Nielsen equivalence in a class of random groups" (joint with Richard Weidmann), Journal of Topology 9 (2016), no. 2, 502-534

  21. "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
  22. "Dynamics on free-by-cyclic groups" , (joint with Spencer Dowdall and Christopher J. Leininger); Geometry and Topology 19 (2015), 2801-2899
  23. "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
  24. "Cannon-Thurston fibers for iwip automorphisms of $F_N$", (joint with Martin Lustig), Jour. London Math. Soc. 91 (2015), no. 1, 203-224
  25. "Unbounded asymmetry of stretch factors", (joint with Spencer Dowdall and Christopher J. Leininger); Comptes Rendus Mathematique, 352 (2014), no. 11, 885-887
  26. "Algorithmic detectability of iwip automorphisms", Bull. London Math. Soc. 46 (2014), no. 2, pp. 279-290; online version at the BLMS site available here
  27. "On hyperbolicity of free splitting and free factor complexes", (joint with Kasra Rafi), Groups, Geometry and Dynamics, 8 (2014), no. 2, pp. 391-414
  28. "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
  29. "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
  30. "Subset currents on free groups", (joint with Tatiana Nagnibeda), Geometriae Dedicata, 166 (2013), 307-348
  31. "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. doi: 10.1093/qmath/hat056
  32. "Random length-spectrum rigidity for free groups", Proceedings of AMS 140 (2012), no. 5, pp. 1549-1560
  33. "Stabilizers of R-trees with free isometric actions of FN ", (with Martin Lustig), Journal of Group Theory, 14 (2011), no. 5, pp. 673-694
  34. "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
  35. "On mathematical contributions of Paul E. Schupp", Illinois J. Math., 54 (2010), no. 1, pp. 1-9, special issue dedicated to Paul Schupp
  36. "Ping-pong and Outer space", (with Martin Lustig), Journal of Topology and Analysis 2 (2010), no. 2, pp. 173-201
  37. "Nielsen equivalence in small cancellation groups", (with Richard Weidmann), preprint
  38. "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
  39. "Intersection form, laminations and currents on free groups ", (with Martin Lustig), Geom. Funct. Anal. 19 (2010), no. 5, pp. 1426-1467
  40. "Geometric Intersection Number and analogues of the Curve Complex for free groups", (with Martin Lustig), Geometry and Topology 13 (2009), pp. 1805-1833
  41. "Some Quantitative Aspects of Fractional Computability", (with Paul Schupp), preprint
  42. "On group-theoretic models of randomness and genericity" (with Paul Schupp), Groups, Geometry and Dynamics 2 (2008), no. 3, pp. 383-404
  43. "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
  44. "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
  45. "On the absence of McShane-type identities for the outer space" (with Igor Rivin), Journal of Algebra 320 (2008), no. 10, pp. 3659-3670
  46. "Clusters, currents and Whitehead's algorithm", Experimental Mathematics 16 (2007), no. 1, pp. 67-76
  47. " 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
  48. "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
  49. "Translation equivalence in free groups", (with Gilbert Levitt, Paul Schupp and Vladimir Shpilrain),  Transact. Amer. Math. Soc. 359 (2007), no. 4, pp. 1527-1546
  50. "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
  51. "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
  52. "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
  53. "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
  54. "Delzant's T-invariant, Kolmogorov complexity and one-relator groups" (with Paul Schupp), Commentari Math. Helv. 80 (2005), no. 4, pp. 911-933
  55. "Kleinian groups and the rank problem", (with Richard Weidmann), Geometry and Topology 9 (2005), pp. 375-402
  56. "Foldings, graphs of groups and the membership problem" (with Richard Weidmann and Alexei Myasnikov), Internat. J. Alg. Comput. 15 (2005), no. 1, 95-128
  57. "Acylindrical accessibility for groups acting on R-trees" (with Richard Weidmann), Math. Zeitschrift 249 (2005), no. 4, pp. 773-782
  58. "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
  59. "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
  60. "Relative hyperbolicity and Artin groups"(with Paul Schupp),  Geom. Dedicata 107 (2004), no. 1, pp. 153-167
  61. "Freely indecomposable groups acting on hyperbolic spaces"(with Richard Weidmann),  Internat. J. Algebra and Comput. 14 (2004), no. 2, pp. 115-171
  62. "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
  63. "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
  64. "Nielsen methods and groups acting on hyperbolic spaces" (with Richard Weidmann), Geometriae Dedicata 98 (2003), no. 1, pp. 95-121
  65. "The Frattini subgroup for subgroups of hyperbolic groups",  J. Group Theory 6 (2003), no. 1, pp. 115-126
  66. "The non-amenability of Schreier graphs for infinite index quasiconvex subgroups of hyperbolic groups", Ensign. Math. 48 (2002), no. 3-4, pp. 359-375
  67. "The geometry of relative Cayley graphs for subgroups of hyperbolic  groups", preprint
  68. "A remark on mapping tori of free group endomorphisms",  preprint
  69. "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
  70. "A note on the Poenaru condition",   J. Group Theory 5 (2002), no. 1,  pp. 119-127
  71. "Stallings foldings and the subgroup structure of free groups" (with Alexey Myasnikov), J. Algebra 248 (2002), no 2,   pp. 608-668, and an Addendum
  72. "The Combination Theorem and Quasiconvexity",  Intern. J. Algebra Comput. 11 (2001), no. 2,  pp. 185-216
  73. "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
  74. "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
  75.  "Mapping tori of endomorphisms of free groups",  Comm. in Algebra 28 (2000), no. 6,  pp. 2895-2917
  76. "Equivalence of some residual properties for word-hyperbolic groups"(with Dani Wise),   J. Algebra 223 (2000), no. 2, pp. 562-583
  77. " Two-generated groups acting on Trees"(with Richard Weidmann), Archiv der Mathematik (Basel) 73 (1999), no. 3, pp. 172-181
  78. "On the structure of two-generated word-hyperbolic groups" (with Richard Weidmann),   Math. Z. 231 (1999), no. 4, pp. 783-801
  79. "One-relator groups and the Howson property", Comm. in Algebra 27 (1999), no. 3,  pp. 1057-1072
  80. "A non-quasiconvexity embedding theorem for hyperbolic groups",   Math Proc. Cambridge Phil. Soc. 127 (1999), no. 3,  pp. 461-486
  81. "Quasiconvexity and amalgams", Intern. J. Alg. Computation 7 (1997), no. 6,  pp. 771-811
  82. "Amalgamated products and the Howson Property" , Canadian Math. Bulletin, 40 (1997), no. 3,  pp. 330-340
  83. "Small cancellation groups and translation numbers", Trans. Amer. Math. Soc., 349 (1997), no. 5,  pp. 1851-1875
  84. "Greenberg's theorem for quasiconvex subgroups of word hyperbolic groups" (with Hamish Short), Canadian J. Math., 48 (1996), no. 6, pp. 1224-1244
  85.  "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
  86. "Detecting quasiconvexity: algorithmic aspects",  AMS DIMACS series, 25,  Geometric and Computational Perspectives on Infinite Groups, 1995, pp. 91-100

My graduate students
  1. Lucas Sabalka (PhD 2006, UIUC); an applied mathematician at Nebraska Global
  2. Gregory Kelsey (PhD 2011, UIUC); Assistant Professor at Bellarmine University
  3. Brent Solie (PhD 2011, UIUC); Assistant Professor at Embry-Riddle Aeronautical University
  4. Patrick Reynolds (PhD 2011, UIUC); Machine learning engineer, Apple
  5. Brian Ray (PhD 2014, UIUC); senior analyst at Daniel H. Wagner Associates)
  6. Neha Gupta (PhD 2016, UIUC); Director of Math Scheduling, Georgia Tech
  7. Caglar Uyanik (PhD 2017, UIUC); Gibbs Assistant Professor, Yale
  8. Elizabeth Field (PhD 2020, UIUC); postdoctoral researcher, University of Utah
  9. Heejoung Kim (UIUC, current PhD student)
  10. Zachary Simon (M.Sc. 2020, City College of CUNY)