Research Interests.

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

Papers.
 

1. "Geometric entropy and Patterson-Sullivan currents", (with Tatiana Nagnibeda), preliminary version
   
2. "Intersection form, laminations and currents on free groups ", (with Martin Lustig), preprint
   
3. "Geometric Intersection Number and analogues of the Curve Complex for free groups", (with Martin Lustig), preprint
   
4. "Some Quantitative Aspects of Fractional Computability", (with Paul Schupp), preprint
   
5. "On group-theoretic models of randomness and genericity" (with Paul Schupp), preprint
   
6. "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
7. "Random quotients of the modular group are rigid and essentially incompressible" (with Paul Schupp), Crelle's Journal, to appear
8. "On the absence of McShane-type identities for the outer space" (with Igor Rivin), preprint
   
9. "Clusters, currents and Whitehead's algorithm", Experimental Mathematics 16 (2007), no. 1, pp. 67-76
   
10. " 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
    
11. "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
    
12. "Translation equivalence in free groups", (with Gilbert Levitt, Paul Schupp and Vladimir Shpilrain),  Transact. Amer. Math. Soc. 359 (2007), no. 4, pp. 1527-1546
13. "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
    
14. "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
15. "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
    
16. "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
    
17. "Delzant's T-invariant, Kolmogorov complexity and one-relator groups" (with Paul Schupp), Commentari Math. Helv. 80 (2005), no. 4, pp. 911-933
    
18. "Kleinian groups and the rank problem", (with Richard Weidmann), Geometry and Topology 9 (2005), pp. 375-402
    
19. "Foldings, graphs of groups and the membership problem" (with Richard Weidmann and Alexei Myasnikov), Internat. J. Alg. Comput. 15 (2005), no. 1, 95-128
20. "Acylindrical accessibility for groups acting on R-trees" (with Richard Weidmann), Math. Zeitschrift 249 (2005), no. 4, pp. 773-782
21. "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
22. "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
23. "Relative hyperbolicity and Artin groups"(with Paul Schupp),  Geom. Dedicata 107 (2004), no. 1, pp. 153-167
24. "Freely indecomposable groups acting on hyperbolic spaces"(with Richard Weidmann),  Internat. J. Algebra and Comput. 14 (2004), no. 2, pp. 115-171
25. "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
26. "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
27. "Nielsen methods and groups acting on hyperbolic spaces" (with Richard Weidmann), Geometriae Dedicata 98 (2003), no. 1, pp. 95-121
28. "The Frattini subgroup for subgroups of hyperbolic groups",  J. Group Theory 6 (2003), no. 1, pp. 115-126
29. "The non-amenability of Schreier graphs for infinite index quasiconvex subgroups of hyperbolic groups", Ensign. Math. 48 (2002), no. 3-4, pp. 359-375
30. "The geometry of relative Cayley graphs for subgroups of hyperbolic  groups", preprint
31. "A remark on mapping tori of free group endomorphisms",  preprint
32. "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
33. "A note on the Poenaru condition",   J. Group Theory 5 (2002), no. 1,  pp. 119-127
34. "Stallings foldings and the subgroup structure of free groups" (with Alexey Myasnikov), J. Algebra 248 (2002), no 2,   pp. 608-668, and an Addendum
35. "The Combination Theorem and Quasiconvexity",  Intern. J. Algebra Comput. 11 (2001), no. 2,  pp. 185-216
36. "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
37. "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
38.  "Mapping tori of endomorphisms of free groups",  Comm. in Algebra 28 (2000), no. 6,  pp. 2895-2917
39. "Equivalence of some residual properties for word-hyperbolic groups"(with Dani Wise),   J. Algebra 223 (2000), no. 2, pp. 562-583
40. " Two-generated groups acting on Trees"(with Richard Weidmann), Archiv der Mathematik (Basel) 73 (1999), no. 3, pp. 172-181
41. "On the structure of two-generated word-hyperbolic groups" (with Richard Weidmann),   Math. Z. 231 (1999), no. 4, pp. 783-801
42. "One-relator groups and the Howson property", Comm. in Algebra 27 (1999), no. 3,  pp. 1057-1072
43. "A non-quasiconvexity embedding theorem for hyperbolic groups",   Math Proc. Cambridge Phil. Soc. 127 (1999), no. 3,  pp. 461-486
44. "Quasiconvexity and amalgams", Intern. J. Alg. Computation 7 (1997), no. 6,  pp. 771-811
45. "Amalgamated products and the Howson Property", Canadian Math. Bulletin, 40 (1997), no. 3,  pp. 330-340
46. "Small cancellation groups and translation numbers", Trans. Amer. Math. Soc., 349 (1997), no. 5,  pp. 1851-1875
47. "Greenberg's theorem for quasiconvex subgroups of word hyperbolic groups" (with Hamish Short), Canadian J. Math., 48 (1996), no. 6, pp. 1224-1244
48.  "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
49. "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:

• Dan Farley
  
• Rob Ghrist
• Kim Whittlesey
• Chris Leininger
• Paul Schupp
• Igor Mineyev
• Sergei Ivanov
• Derek Robinson
  

People:

• 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
• Simon Heller
  

• 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

 

• 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)
  

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