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92. Fully residually free groups and
graphs labeled by infinite words (with
V.Remeslennikov and D.Serbin), to appear in Internat. J. Algebra and Comput. 91. A practical attack on some braid group
based cryptographic protocols (with
V.Shpilrain and A.Ushakov). CRYPTO 2005, Lecture Notes Comp. Sci,
3621 (2005),
86-96.
90. On positive theories of groups with
regular length functions (with B.Khan and
D.Serbin). 89. Algebraic geometry over groups III:
Elements of model theory (with A.Kvaschuk and V. Remeslennikov). J. of Algebra, 288
(2005), no. 1, 78-98. [.pdf] 88. Finitary Andrews-Curtis conjecture (with A.V. Borovik and A. Lubotzky), submitted. We formulate a generalized AC-conjecture for an arbitrary group and prove that it holds for every finite group. In fact, there are no counter examples to the generalized AC conjecture so far. Such examples would shed some light on potential counterexamples to the original AC conjecture. 87. Pattern Recognition and Minimal
Words in Free Groups of Rank 2 (with
Robert M. Haralick and Alexei D. Miasnikov), submitted. [.pdf] We describe a linear time probabilistic algorithm to recognize Whitehead minimal elements (elements of minimal length in their automorphic orbits) in free groups of rank 2. Moreover, for a non-minimal element the algorithm gives an automorphism that will most likely reduce the length of the element. This method is based on linear regression and pattern recognition techniques. I did not believe this myself first, until I've seen it working with my own eyes. 86. Effective JSJ decompositions (with
O.Kharlampovich), Contemp. Math., 378,
Amer.
Math. Soc., 85. Regular free length
functions on Lyndon's free Z[t]-group F^Z[t] (with V.
Remeslennikov and D.Serbin), Contemp.
Math., 378, Amer. Math. Soc.,
RI, 2005, 37-77. [.pdf] 84. The conjugacy problem in
amalgamated products I: regular elements and black holes (with A. Borovik and V. Remeslennikov). Preprint. 83. The conjugacy problem in
amalgamated products II: generic complexity (with A. Borovik and V. Remeslennikov). Preprint. 82. Big powers and free
constructions (with A.Kvaschuk), submitted. 81. Algorithmic
stratification of the conjugacy problem in Miller’s groups (with A. Borovik and V. Remeslennikov). Preprint. [.pdf] 80. Groups whose universal
theory is axiomatizable by quasi-identities (with B.Fine, A.M.Gaglione and D.Spellman), to appear in J. of Group Theory. 79. Algebraic geometry over
free groups: lifting solutions into generic points (with O.
Kharlampovich), Contemp. Math.,
378, Amer. Math. Soc.,
318. [.pdf] 78. Implicit function
theorems over free groups (with O.Kharlampovich). J. of Algebra, 290
(2005), no. 1, 1-203. [.pdf] 77. Quotient tests and
Gröbner bases (with M. Kreuzer, G. Rosenberger and A. Ushakov), submitted.
[.pdf]
76. The halting problem is
almost always decidable (with J. Hamkins), Preprint. [.pdf] 75. Quotient tests and
random walks in computational group theory (with A. Borovik), submitted.
[.pdf] 74. The conjugacy problem in
HNN-extensions I: regular elements, black holes and generic complexity (with A. Borovik and V.
Remeslennikov). Preprint. 73. Isomorphism problem for
finitely generated fully residually free groups (with I. Bumagin and O. Kharlampovich), submitted.[.pdf] 72. Exponential extensions
of groups (with O. Kharlampovich, V. Remeslennikov and D. Serbin), to appear in J. of Group Theory. [.pdf] 71. One variable equations
over hyperbolic groups (with A. Kvaschuk). Preprint. 70. On the Hanna Newmann
Conjecture (with T.Jitsukawa and B.Khan). Preprint.
[.pdf] 69. Elementary theory of
free non-abelian groups. The Tarski problem. (with O.Kharlampovich). Preprint. 68. Heuristics for Whitehead
Minimization Problem (with R.M.Haralick and A.D. Miasnikov). Preprint. 67. Theorems on hierarchy in
generalized recursion theory (with V. Belyaev). Preprint. 66. Average-case complexity
and decision problems in group theory (with I.Kapovich, P.Schupp and V.Shpilrain), Adv. Math.
190 (2005),
no.2, 190-205. [.pdf] 65. Subgroups of fully residually
free groups: algorithmic problems (with O. Kharlampovich, V. Remeslennikov and D.
Serbin). Group theory, statistics, and
cryptography, 63-101, Contemp. Math., 360,
Amer. Math. Soc., 64. Pattern Recognition Approaches to
Solving Combinatorial Problems in Free Groups
(with Robert M. Haralick and Alexei D. Miasnikov), Contemp. Math., 349,
Amer. Math. Soc., In this survey paper we present some pattern recognition techniques that can be used in studying free groups. As an illustration we apply these methods to Whitehead minimization problem. 63. Whitehead method and Genetic
Algorithms (with Alexei D. Miasnikov)
Contemp.
Math., 349, Amer. Math. Soc., We describe a genetic Whitehead's algorithm for finding minimal elements (elements of minimal length in automorphic orbits) in arbitrary free groups. The classical Whitehead's algorithm is extremely inefective for free groups of big ranks, but this genetic version of it works all right. Also we formulate several mathematical conjectures that came out of our experiments. 62. Discriminating and
co-discriminating groups (with P. Shumyatsky). J. Group Theory 7 (2004), no. 1, 135–142. [.pdf] 61. One Variable Equations
in free groups via context free languages (with R. Gilman). Computational and experimental group theory, 83-88, Contemp. Math., 349, Amer. Math. Soc., 60. Elementary theory of
free groups (with B.Fine, A.M.Gaglione and D.Spellman), Proc. Of the Oxford/St. Andrews Conf. on
Combinatorial Group Theory, Note Ser. 304 (2003), 197-231. 59. Foldings, graphs of
groups and the membership problem (with I.Kapovich and R.Weidmann), Internat. J. Algebra Comput. 13 (2005), no. 1, 95-128. [.pdf] 58. Multiplicative measures
on free groups (with A. Borovik and V. Remeslennikov). Internat. J. Algebra Comput. 13 (2003),
no. 6, 705-731. [.pdf] 57. The Andrews-Curtis
Conjecture and Black Box Groups (with A.V. Borovik and E.I. Khukhro). Internat. J. Algebra Comput.,
13 (2003),
no. 4, 415-436. [.pdf] We introduce Andrews-Curtis graphs of groups and discuss their connection with black-box groups and the replacement algorithm, study AC conjecture for finite groups, and describe possible attacks on AC conjecture via finite groups. 56. Automorphic orbits in
free groups (with V.Shpilrain). J.
of Algebra, 269 (2003), no. 1, 18-27. [.pdf] 55. Generic-case complexity
and decision problems in group theory (with Schupp and V.Shpilrain). J. of Algebra, 264 (2003), 665-694. [.pdf] 54. Discriminating
completions of hyperbolic groups (with G. Baumslag and V. Remeslennikov). Geom. Dedicata 92 (2002), 115-143. [.pdf] 53. Measuring sets in infinite
groups (with A. Borovik and V. Shpilrain). Computational and statistical group theory ( Amer. Math. Soc., 52. Open problems in
combinatorial group theory. Second edition (with G. Baumslag and V. Shpilrain). Combinatorial and geometric
group theory (New York, 2000/Hoboken, NJ, 2001), 1-38, Contemp. Math., 296,
Amer. Math. Soc., 51. On the Andrews-Curtis
equivalence (with A.D. Miasnikov and V. Shpilrain). Combinatorial
and geometric group theory (New York, 2000/Hoboken, NJ, 2001), 183-198. Contemp.
Math., 296,
Amer. Math. Soc., We construct various examples of balanced presentations of the trivial group and show that some of them satisfy AC conjecture. We also consider AC-equivalence in metabelian groups and reveal some interesting connections of it with well-known problems in K-theory. 50. Stallings foldings and
subgroups of free groups (with (2002), 608-668. 49. Discriminating groups
(with B. Fine, A.M. Gaglione and D. Spellman). J. of Group Theory 4 (2001), no. 4, 463–474. [.pdf] 48. Implicit function
theorem over free groups and genus problem (with O. Kharlampovich). Knots, braids, and mapping class groups---papers dedicated to
Joan S. Birman (New York, 1998), 77-83, AMS/IP Stud. Adv.
Math., 24, Amer. Math. Soc., 47. Balanced presentations
of the trivial group on two generators and the Andrews-Curtis conjecture (with A.D. Miasnikov). Groups and computation, III (Columbus, OH, 1999), 257-263.
Ohio State Univ. Math. Res. Inst. Publ. 8, We proved that every balanced presentation on two generators of the trivial group satisfies the AC conjecture. 46. Discriminating and
co-discriminating groups (with G. Baumslag and V. Remeslennikov). J. Group Theory 3 (2000), no. 4, 467–479. [.pdf] 45. Algebraic geometry 2: logical
foundations (with V. Remeslennikov). J.
of Algebra 234 (2000),
225-276. [.pdf] 44. Centroids of groups
(with 43.
Algebraic geometry over groups (with
G. Baumslag and V. Remeslennikov). Algorithmic problems in groups and semigroups (Lincoln, NE, 1998), 35-50, Trends Math., Birkhäuser 42. Aspects of the theory of
free groups (with K. Bencsath, B. Fine, A. Gaglione, G. Rosenberger and D. Spellman). Algorithmic problems in groups and semigroups ( NE, 1998), 35--50, Trends Math., Birkhäuser
Boston, 41. Malnormality is
decidable in free groups (with G.Baumslag and V.Remeslennikov). Internat. Journal Algebra Comput. 9 (1999), no.6, 687-692. [.pdf] 40. Open problems in
combinatorial group theory (with G. Baumslag and V. Shpilrain). Groups, languages and geometry (South Hadley, MA,
1998), 1-27,
Contemp. Math., 250, Amer. Math. Soc., 39. Equations over
Q-completions of hyperbolic groups (with O. Kharlampovich and E. Lyutikova). Trans. Amer. Math. Soc. 351 (1999),
no. 7, 2961-2978. 38. Algebraic geometry over and V. Remeslennikov). J. of Algebra 219 (1999), 16-79. [.pdf] 37. Description of fully
residually free groups and irreducible affine varieties over a free group (with O. Kharlampovich). Summer School in Group Theory in CRM
Proc. Lecture Notes, 17, Amer. Math. Soc., 36. A classification of fully residually free groups of rank three
or less (with A. Gaglione, B. Fine, G. Rosenberger and D.Spellman). J. of Algebra, 200 (1998), no. 2, 571-605. 35. Irreducible affine
varieties over a free group. II:
Systems in triangular quasi-quadratic form and description of residually free
groups (with O. Kharlampovich). J.
of Algebra 200
(1998), no. 2, 517--570. [.pdf] 34. Irreducible affine
varieties over a free group. I:
Irreducibility of quadratic equations and Nullstellensatz (with O.
Kharlampovich). J. of Algebra 200 (1998), no. 2, 472-516. 33. Equations in free a
Q-group (with O. Kharlampovich). Trans. Amer. Math. Soc. 350 (1998), no. 3, 947-974. [.pdf] 32. Hyperbolic groups and
free constructions (with O. Kharlampovich). Trans. Amer. Math. Soc. 350 (1998),
no. 2, 571-613. [.pdf] 31. Two theorems about
equationally Noetherian groups (with G. Baumslag and V. Romankov). J, of Algebra 194 (1997),
654-613. [.pdf] 30. Formal power series
representations of free exponential groups (A.M. Gaglione, V.N. Remeslennikov, D. Spellman). Comm. Algebra, 25
(1997), no. 2, 631-648. 29. Exponential groups 2: extension
of centralizers and tensor completion of CSA-groups (with V. Remeslennikov). Internat. J. Algebra Comput. 6 (1996),
no. 6, 687-711. [.pdf] 28. Length functions on free
exponential groups (with V. Remeslennikov). Proc. IITPM SO RAN 26, 27. Length functions on free
exponential groups (with V. Remeslennikov). Proc.of Intern. Conference in Analysis and
Geometry, 26. CSA groups and separated
free constructions (with D. Gildenhuys and O. Kharlampovich). Bull. Austr. Math. Soc. 52 (1995), no. 1, 63-84. [.pdf] 25. Logical languages and
axioms for groups with a length function (with O. Kharlampovich and V. Remeslennikov). Proc. Inst. Appl. Math.
Russian Acad. Sci. 20 (1995), 1-8. 24. Exponential groups I:
foundations of the theory and tensor completions (with V. Remeslennikov). Siberian Math. J.. 35 (1994),
no. 5, 1106-1118. 23. Approaches to the theory
of computability (with Logika, 1994, v.4. 22. Recursively enumerable
degrees over the reals (with V. Belyaev). Proc. of Intern. Algebraic Conf.,
21. Model-theoretic problems
of algebra. Dissertation, VINITI,
20. Admissible sets in group
theory (with V.Remeslennikov). Algebra
i logika, 3 (1992).
19. Weak second order logic
in group theory (with V.Remeslennikov). Contemp. Math. 131 (1992), no. 2, 273-277.
18. Logic of finite types in
group theory (with V.Remeslennikov). Bull.
Siberian Math. Soc.,
17. Model theory of bilinear mappings. Siberian
J. Math., 31 (1990), no. 3, 94-108. 16. Invariants of bilinear
mappings. Siberian J. Math., 31 (1990), no. 1, 104-115. 15. Model-theoretical
problems in group theory (with V.Remeslennikov). Problems of Algebra
4, 14. Modules over local ring.
Siberian J. Math., 30 (1989), no. 3, 72-83. 13. Structure of models and
criteria of decidability of complete theories of finite- dimensional algebras (with
V.Remeslennikov). Izvest. Akademii
Nauk USSR, ser.math 53 (1989), no. 2., 379-397. 12. Recursive p-adic numbers
and elementary theories of finitely generated pro-p-groups (with V.Remeslennikov). Izvest. Akademii Nauk USSR, ser.math 30 (1988), no.3. 11. Abstract isomorphisms
and elementary theories of finite dimensional algebras and unipotent groups. Doklady Akademii Nauk 10. Free operator groups
with length function and finitely generated group of operators. Computable
invariants of algebraic structures., 9. Elementary group equivalence
with the integral length function (with V.Remeslennikov). 8. Stable torsion-free rings
and nilpotent groups. VII 7. Generalized Nielsen
transformations and trivial groups. Math.
Notes, 5 (1984), no. 4, 491- 495. 6. Finite-dimensional
algebras and k-groups of finite rank. Contemp.
Math., 33 (1984), 436- 453. 5. Definability of Mal'cev
bases and elementary properties of finite dimensional algebras. II (with V.Remeslennikov). Siberian J. Math.,
24 (1982), no. 2, 97-113. 4. Definability of Mal'cev
bases and elementary properties of finite dimensional algebras. I (with V.Remeslennikov). Siberian J. Math.,
23 (1982), no. 5, 152-167. 3. Elementary properties of
powered nilpotent groups (with V.Remeslennikov). Doklady Akademii Nayuk 258 (1981), no.
5, 1056-1059. 2. Isomorphism and
elementary properties of powered nilpotent groups (with V.Remeslennikov). Proc.Math.Institute 2, Nauka, 1. On the approximability of
groups of outer automorphisms of free groups of finite rank. Algebra and Logic, 20 (1981), no.3, 291-299.
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