Research Interests.



    My mathematical interests are:
During the period of 2010-2015 I was a member of the Editorial Board for the LMS Journal of Computation and Mathematics. This was a free (also known as "diamond open access") electronic journal published by the London Mathematical Society. Unlike most other open access journals, the journal did not charge any page or publication fees to the authors, so the journal was free both for the authors and for the readers.


I am a Fellow of the American Mathematical Society.


My reasonably up-to-date CV (September 22, 2016)

I am a co-organizer, with Chris Leininger and Steve Bdradlow, of the Geometry, Groups and Dynamics/GEAR Seminar at UIUC.

See the schedule of the GGD/GEAR seminar here, and a web-page of this seminar at the GEAR website here.

Most of the talks in the GGD/GEAR seminar are available on YouTube here.

I am co-organizing, with Alexei Myasnikov, Bob Gilman and Carl Jokusch, a conference "Groups and Computation: interaction between geometric group theory, computability and computer science", Stevens Institute of Technology, June 26-30, 2017. The conference is dedicated to 80th birthday of Paul Schupp.

See the poster for the conference here.



Non-mathematical writing.
 


Video




Papers.


Link to my papers posted at arxiv.og 


  1. "Detecting fully irreducible automorphisms: a polynomial time algorithm. With an appendix by Mark C. Bell", preprint
  2. "Endomorphisms, train track maps, and fully irreducible monodromies ", (joint with Spencer Dowdall and Christopher Leininger), preprint
  3. "Cannon-Thurston maps for hyperbolic free group extensions", (joint with Spencer Dowdall and Samuel Taylor), to appear in Israel J. Math.
  4. "Musings on generic-case complexity", preprint
  5. "On purely loxodromic actions" , Monatshefte für Mathematik 181 (2016), no. 1, pp. 89-101
  6. "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), preprint
  7. "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
  8. "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), to appear
  9. "Nielsen equivalence in a class of random groups" (joint with Richard Weidmann), Journal of Topology 9 (2016), no. 2, 502-534
  10. "An integral weight realization theorem for subset currents on free groups" , Topology Proceedings, to appear
  11. "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
  12. "Dynamics on free-by-cyclic groups" , (joint with Spencer Dowdall and Christopher J. Leininger); Geometry and Topology 19 (2015), 2801-2899
  13. "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
  14. "Cannon-Thurston fibers for iwip automorphisms of $F_N$", (joint with Martin Lustig), Jour. London Math. Soc. 91 (2015), no. 1, 203-224
  15. "Unbounded asymmetry of stretch factors", (joint with Spencer Dowdall and Christopher J. Leininger); Comptes Rendus Mathématique, 352 (2014), no. 11, 885-887
  16. "Algorithmic detectability of iwip automorphisms", Bull. London Math. Soc. 46 (2014), no. 2, pp. 279-290; online version at the BLMS site available here
  17. "On hyperbolicity of free splitting and free factor complexes", (joint with Kasra Rafi), Groups, Geometry and Dynamics, 8 (2014), no. 2, pp. 391-414
  18. "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
  19. "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
  20. "Subset currents on free groups", (joint with Tatiana Nagnibeda), Geometriae Dedicata, 166 (2013), 307-348
  21. "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
  22. "Random length-spectrum rigidity for free groups", Proceedings of AMS 140 (2012), no. 5, pp. 1549-1560
  23. "Stabilizers of R-trees with free isometric actions of FN ", (with Martin Lustig), Journal of Group Theory, 14 (2011), no. 5, pp. 673-694
  24. "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
  25. "On mathematical contributions of Paul E. Schupp", Illinois J. Math., 54 (2010), no. 1, pp. 1-9, special issue dedicated to Paul Schupp
  26. "Ping-pong and Outer space", (with Martin Lustig), Journal of Topology and Analysis 2 (2010), no. 2, pp. 173-201
  27. "Nielsen equivalence in small cancellation groups", (with Richard Weidmann), preprint
  28. "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
  29. "Intersection form, laminations and currents on free groups ", (with Martin Lustig), Geom. Funct. Anal. 19 (2010), no. 5, pp. 1426-1467
  30. "Geometric Intersection Number and analogues of the Curve Complex for free groups", (with Martin Lustig), Geometry and Topology 13 (2009), pp. 1805-1833
  31. "Some Quantitative Aspects of Fractional Computability", (with Paul Schupp), preprint
  32. "On group-theoretic models of randomness and genericity" (with Paul Schupp), Groups, Geometry and Dynamics 2 (2008), no. 3, pp. 383-404
  33. "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
  34. "Random quotients of the modular group are rigid and essentially incompressible" (with Paul Schupp), Journal f�r die reine und angewandte Mathematik (Crelle's Journal), 628 (2009),  pp. 91�119
  35. "On the absence of McShane-type identities for the outer space" (with Igor Rivin), Journal of Algebra 320 (2008), no. 10, pp. 3659-3670
  36. "Clusters, currents and Whitehead's algorithm", Experimental Mathematics 16 (2007), no. 1, pp. 67-76
  37. " 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
  38. "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
  39. "Translation equivalence in free groups", (with Gilbert Levitt, Paul Schupp and Vladimir Shpilrain),  Transact. Amer. Math. Soc. 359 (2007), no. 4, pp. 1527-1546
  40. "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
  41. "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
  42. "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
  43. "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
  44. "Delzant's T-invariant, Kolmogorov complexity and one-relator groups" (with Paul Schupp), Commentari Math. Helv. 80 (2005), no. 4, pp. 911-933
  45. "Kleinian groups and the rank problem", (with Richard Weidmann), Geometry and Topology 9 (2005), pp. 375-402
  46. "Foldings, graphs of groups and the membership problem" (with Richard Weidmann and Alexei Myasnikov), Internat. J. Alg. Comput. 15 (2005), no. 1, 95-128
  47. "Acylindrical accessibility for groups acting on R-trees" (with Richard Weidmann), Math. Zeitschrift 249 (2005), no. 4, pp. 773-782
  48. "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
  49. "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
  50. "Relative hyperbolicity and Artin groups"(with Paul Schupp),  Geom. Dedicata 107 (2004), no. 1, pp. 153-167
  51. "Freely indecomposable groups acting on hyperbolic spaces"(with Richard Weidmann),  Internat. J. Algebra and Comput. 14 (2004), no. 2, pp. 115-171
  52. "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
  53. "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
  54. "Nielsen methods and groups acting on hyperbolic spaces" (with Richard Weidmann), Geometriae Dedicata 98 (2003), no. 1, pp. 95-121
  55. "The Frattini subgroup for subgroups of hyperbolic groups",  J. Group Theory 6 (2003), no. 1, pp. 115-126
  56. "The non-amenability of Schreier graphs for infinite index quasiconvex subgroups of hyperbolic groups", Ensign. Math. 48 (2002), no. 3-4, pp. 359-375
  57. "The geometry of relative Cayley graphs for subgroups of hyperbolic  groups", preprint
  58. "A remark on mapping tori of free group endomorphisms",  preprint
  59. "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
  60. "A note on the Poenaru condition",   J. Group Theory 5 (2002), no. 1,  pp. 119-127
  61. "Stallings foldings and the subgroup structure of free groups" (with Alexey Myasnikov), J. Algebra 248 (2002), no 2,   pp. 608-668, and an Addendum
  62. "The Combination Theorem and Quasiconvexity",  Intern. J. Algebra Comput. 11 (2001), no. 2,  pp. 185-216
  63. "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
  64. "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
  65.  "Mapping tori of endomorphisms of free groups",  Comm. in Algebra 28 (2000), no. 6,  pp. 2895-2917
  66. "Equivalence of some residual properties for word-hyperbolic groups"(with Dani Wise),   J. Algebra 223 (2000), no. 2, pp. 562-583
  67. " Two-generated groups acting on Trees"(with Richard Weidmann), Archiv der Mathematik (Basel) 73 (1999), no. 3, pp. 172-181
  68. "On the structure of two-generated word-hyperbolic groups" (with Richard Weidmann),   Math. Z. 231 (1999), no. 4, pp. 783-801
  69. "One-relator groups and the Howson property", Comm. in Algebra 27 (1999), no. 3,  pp. 1057-1072
  70. "A non-quasiconvexity embedding theorem for hyperbolic groups",   Math Proc. Cambridge Phil. Soc. 127 (1999), no. 3,  pp. 461-486
  71. "Quasiconvexity and amalgams", Intern. J. Alg. Computation 7 (1997), no. 6,  pp. 771-811
  72. "Amalgamated products and the Howson Property" , Canadian Math. Bulletin, 40 (1997), no. 3,  pp. 330-340
  73. "Small cancellation groups and translation numbers", Trans. Amer. Math. Soc., 349 (1997), no. 5,  pp. 1851-1875
  74. "Greenberg's theorem for quasiconvex subgroups of word hyperbolic groups" (with Hamish Short), Canadian J. Math., 48 (1996), no. 6, pp. 1224-1244
  75.  "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
  76. "Detecting quasiconvexity: algorithmic aspects",  AMS DIMACS series, 25,  Geometric and Computational Perspectives on Infinite Groups, 1995, pp. 91-100

   Miscellaneous files


   Group Theory links.

Geometric Group Theory at UIUC:

People:


My graduate students:

Preprint servers:

 
Conferences:
 
Miscelaneous  Group Theory resources: Acknowledgement of support:
   
      My research has been supported by the National Science Foundation grants DMS-0404991 (2004-2007),DMS-0603921 (2006-2009) DMS-0904200 (2009-2013) and the Collaboration Grant
no. 279836 (2013-2014) from the Simons Foundation , and is currently supported by the National Science Foundation grant DMS-1405146 (2014-2017)

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