Research Interests.



    My mathematical interests are:
A brief summary of my recent work (for about the last five years) is available here, January 2017

My reasonably up-to-date CV (September 6, 2017)

I am a Fellow of the American Mathematical Society.

Editorial work



Other professional service




Non-mathematical writing.
 


Video




Papers.


Link to my papers posted at arxiv.og 


  1. "Couting conjugacy classes in Out(FN)", (joint with Michael Hull), preprint
  2. "Stable Strata of Geodesics in Outer Space", (joint with Yael Algom-Kfir and Catherine Pfaff), preprint
  3. "Detecting fully irreducible automorphisms: a polynomial time algorithm. With an appendix by Mark C. Bell", Experimental Mathematics, to appear
  4. "Endomorphisms, train track maps, and fully irreducible monodromies ", (joint with Spencer Dowdall and Christopher Leininger), Groups, Geometry, and Dynamics; to appear
  5. "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), Math. Proc. Cambr. Phil. Soc., to appear
  6. "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
  7. "An integral weight realization theorem for subset currents on free groups" , Topology Proceedings 50 (2017), pp. 213–236
  8. "Wolfgang Haken: A biographical sketch", Illinois Journal of Mathematics 60 (2016), no. 1, pp. 3-9; special issue in honor of Wolfgang Haken
  9. "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
  10. "Musings on generic-case complexity", preprint
  11. "On purely loxodromic actions" , Monatshefte für Mathematik 181 (2016), no. 1, pp. 89-101
  12. "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
  13. "Nielsen equivalence in a class of random groups" (joint with Richard Weidmann), Journal of Topology 9 (2016), no. 2, 502-534

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

 
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 grants DMS-1405146 (2014-2018) and DMS-1710868 (2017-2020)

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