Research Interests.

•   Geometric and Combinatorial Group Theory
•   R-trees, Outer space and Out(F_n)
•   3-dimensional Topology and Kleinian Groups
•   Manifolds of Non-positive Curvature and CAT(0)-spaces
•   Logic, Computational and Algorithmic aspects of Algebra

Non-mathematical writing.
 

• "The dangers of the ``author pays'' model in mathematical publishing", Notices of the American Mathematical Society, to appear

Papers.
 

1. "Spectral rigidity of automorphic orbits in free groups", (joint with Mathieu Carette, Stefano Francaviglia and Armando Martino), preprint
   
2. "Generalized geodesic currents on free groups", (joint with Tatiana Nagnibeda), preprint
3. "Invariant laminations for irreducible automorphisms of free groups", (joint with Martin Lustig), preprint
4. "Random length-spectrum rigidity for free groups", Proceedings of AMS, to appear
5. "Stabilizers of R-trees with free isometric actions of F_N ", (with Martin Lustig), Journal of Group Theory, 14 (2011), no. 5, pp. 673–694
6. "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
   
7. "On mathematical contributions of Paul E. Schupp", Illinois J. Math., 54 (2010), no. 1, pp. 1-9, special issue dedicated to Paul Schupp
   
8. "Ping-pong and Outer space", (with Martin Lustig), Journal of Topology and Analysis 2 (2010), no. 2, pp. 173-201
   
9. "Nielsen equivalence in small cancellation groups", (with Richard Weidmann), preprint
   
10. "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
    
11. "Intersection form, laminations and currents on free groups ", (with Martin Lustig), Geom. Funct. Anal. 19 (2010), no. 5, pp. 1426-1467
12. "Geometric Intersection Number and analogues of the Curve Complex for free groups", (with Martin Lustig), Geometry and Topology 13 (2009), pp. 1805-1833
    
13. "Some Quantitative Aspects of Fractional Computability", (with Paul Schupp), preprint
    
14. "On group-theoretic models of randomness and genericity" (with Paul Schupp), Groups, Geometry and Dynamics 2 (2008), no. 3, pp. 383-404
15. "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
16. "Random quotients of the modular group are rigid and essentially incompressible" (with Paul Schupp), Journal für die reine und angewandte Mathematik (Crelle's Journal), 628 (2009),  pp. 91–119
17. "On the absence of McShane-type identities for the outer space" (with Igor Rivin), Journal of Algebra 320 (2008), no. 10, pp. 3659-3670
18. "Clusters, currents and Whitehead's algorithm", Experimental Mathematics 16 (2007), no. 1, pp. 67-76
    
19. " 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
    
20. "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
    
21. "Translation equivalence in free groups", (with Gilbert Levitt, Paul Schupp and Vladimir Shpilrain),  Transact. Amer. Math. Soc. 359 (2007), no. 4, pp. 1527-1546
22. "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
    
23. "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
24. "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
    
25. "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
    
26. "Delzant's T-invariant, Kolmogorov complexity and one-relator groups" (with Paul Schupp), Commentari Math. Helv. 80 (2005), no. 4, pp. 911-933
    
27. "Kleinian groups and the rank problem", (with Richard Weidmann), Geometry and Topology 9 (2005), pp. 375-402
    
28. "Foldings, graphs of groups and the membership problem" (with Richard Weidmann and Alexei Myasnikov), Internat. J. Alg. Comput. 15 (2005), no. 1, 95-128
29. "Acylindrical accessibility for groups acting on R-trees" (with Richard Weidmann), Math. Zeitschrift 249 (2005), no. 4, pp. 773-782
30. "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
31. "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
32. "Relative hyperbolicity and Artin groups"(with Paul Schupp),  Geom. Dedicata 107 (2004), no. 1, pp. 153-167
33. "Freely indecomposable groups acting on hyperbolic spaces"(with Richard Weidmann),  Internat. J. Algebra and Comput. 14 (2004), no. 2, pp. 115-171
34. "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
35. "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
36. "Nielsen methods and groups acting on hyperbolic spaces" (with Richard Weidmann), Geometriae Dedicata 98 (2003), no. 1, pp. 95-121
37. "The Frattini subgroup for subgroups of hyperbolic groups",  J. Group Theory 6 (2003), no. 1, pp. 115-126
38. "The non-amenability of Schreier graphs for infinite index quasiconvex subgroups of hyperbolic groups", Ensign. Math. 48 (2002), no. 3-4, pp. 359-375
39. "The geometry of relative Cayley graphs for subgroups of hyperbolic  groups", preprint
40. "A remark on mapping tori of free group endomorphisms",  preprint
41. "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
42. "A note on the Poenaru condition",   J. Group Theory 5 (2002), no. 1,  pp. 119-127
43. "Stallings foldings and the subgroup structure of free groups" (with Alexey Myasnikov), J. Algebra 248 (2002), no 2,   pp. 608-668, and an Addendum
44. "The Combination Theorem and Quasiconvexity",  Intern. J. Algebra Comput. 11 (2001), no. 2,  pp. 185-216
45. "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
46. "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
47.  "Mapping tori of endomorphisms of free groups",  Comm. in Algebra 28 (2000), no. 6,  pp. 2895-2917
48. "Equivalence of some residual properties for word-hyperbolic groups"(with Dani Wise),   J. Algebra 223 (2000), no. 2, pp. 562-583
49. " Two-generated groups acting on Trees"(with Richard Weidmann), Archiv der Mathematik (Basel) 73 (1999), no. 3, pp. 172-181
50. "On the structure of two-generated word-hyperbolic groups" (with Richard Weidmann),   Math. Z. 231 (1999), no. 4, pp. 783-801
51. "One-relator groups and the Howson property", Comm. in Algebra 27 (1999), no. 3,  pp. 1057-1072
52. "A non-quasiconvexity embedding theorem for hyperbolic groups",   Math Proc. Cambridge Phil. Soc. 127 (1999), no. 3,  pp. 461-486
53. "Quasiconvexity and amalgams", Intern. J. Alg. Computation 7 (1997), no. 6,  pp. 771-811
54. "Amalgamated products and the Howson Property", Canadian Math. Bulletin, 40 (1997), no. 3,  pp. 330-340
55. "Small cancellation groups and translation numbers", Trans. Amer. Math. Soc., 349 (1997), no. 5,  pp. 1851-1875
56. "Greenberg's theorem for quasiconvex subgroups of word hyperbolic groups" (with Hamish Short), Canadian J. Math., 48 (1996), no. 6, pp. 1224-1244
57.  "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
58. "Detecting quasiconvexity: algorithmic aspects",  AMS DIMACS series, 25,  Geometric and Computational Perspectives on Infinite Groups, 1995, pp. 91-100

   Group Theory links.

Geometric Group Theory at UIUC:

• Kim Whittlesey
• Chris Leininger
• Paul Schupp
• Igor Mineyev
• Sergei Ivanov
• Derek Robinson
• Jayadev Athreya
• Nathan Dunfield
  

People:

• Martin Lustig (my co-author)
  
• Richard Weidmann  (my co-author)
  
• Steve Gersten
  
• Misha Kapovich  (my older brother)
  
• Mladen Bestvina
  
• Dani Wise  (my co-author)
  
• Alexei Myasnikov (my co-author)
  
• Vladimir Shpilrain  (my co-author)
  
• Hamish Short (my co-author)
  
• Tatiana Smirnova-Nagnibeda (my co-author)
  
• Vadim Kaimanovich (my co-author)
  
• Gilbert Levitt (my co-author)
  
• Igor Rivin (my co-author)
• Peter Brinkmann (a former UIUC postdoc)
• Patrick Bahls (a former UIUC postdoc)
• List of web pages of Geometric Group theorists (at "The World of Groups")
• Jon Mccammond's list of people in Geometric Group Theory

• Lucas Sabalka (currently a postdoc at SUNY Binghamton)
  
• Greg Kelsey
• Brent Solie
• Patrick Reynolds
  

• Group Theory Preprint Archive at xxx.lanl.gov
• University of Southhampton preprint server

• Albany Group Theory Conference
• Geometric Groups on the Gulf Coast (G3) conference
  
• Wasatch Topology Conference
• Cornell Topology Festival

 

• John Mccammond's Geometric Group Theory page
  
• MAGNUS Computational Group Theory Software Project, CUNY
• New York Group Theory Seminar
• Group Pub Forum
• Group Theory Problem List of Mladen Bestvina
• Open Problems in Combinatorial Group Theory (at CUNY)
  

Back to Ilya Kapovich's homepage