Groups with free regular length functions in $\mathbb {Z}^n$
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- by Olga Kharlampovich, Alexei Myasnikov, Vladimir Remeslennikov and Denis Serbin PDF
- Trans. Amer. Math. Soc. 364 (2012), 2847-2882 Request permission
Abstract:
This is the first paper in a series of three where we take on the unified theory of non-Archimedean group actions, length functions and infinite words. Our main goal is to show that group actions on $\mathbb {Z}^n$-trees give one a powerful tool to study groups. All finitely generated groups acting freely on $\mathbb {R}$-trees also act freely on some $\mathbb {Z}^n$-trees, but the latter ones form a much larger class. The natural effectiveness of all constructions for $\mathbb {Z}^n$-actions (which is not the case for $\mathbb {R}$-trees) comes along with a robust algorithmic theory. In this paper we describe the algebraic structure of finitely generated groups acting freely and regularly on $\mathbb {Z}^n$-trees and give necessary and sufficient conditions for such actions.References
- Roger Alperin and Hyman Bass, Length functions of group actions on $\Lambda$-trees, Combinatorial group theory and topology (Alta, Utah, 1984) Ann. of Math. Stud., vol. 111, Princeton Univ. Press, Princeton, NJ, 1987, pp. 265–378. MR 895622
- Roger C. Alperin and Kenneth N. Moss, Complete trees for groups with a real-valued length function, J. London Math. Soc. (2) 31 (1985), no. 1, 55–68. MR 810562, DOI 10.1112/jlms/s2-31.1.55
- Hyman Bass, Group actions on non-Archimedean trees, Arboreal group theory (Berkeley, CA, 1988) Math. Sci. Res. Inst. Publ., vol. 19, Springer, New York, 1991, pp. 69–131. MR 1105330, DOI 10.1007/978-1-4612-3142-4_{3}
- Mladen Bestvina and Mark Feighn, Stable actions of groups on real trees, Invent. Math. 121 (1995), no. 2, 287–321. MR 1346208, DOI 10.1007/BF01884300
- I. M. Chiswell, Abstract length functions in groups, Math. Proc. Cambridge Philos. Soc. 80 (1976), no. 3, 451–463. MR 427480, DOI 10.1017/S0305004100053093
- Ian Chiswell, Introduction to $\Lambda$-trees, World Scientific Publishing Co., Inc., River Edge, NJ, 2001. MR 1851337, DOI 10.1142/4495
- I. M. Chiswell, $A$-free groups and tree-free groups, Groups, languages, algorithms, Contemp. Math., vol. 378, Amer. Math. Soc., Providence, RI, 2005, pp. 79–86. MR 2159315, DOI 10.1090/conm/378/07011
- Benjamin Fine, Gerhard Rosenberger, and Michael Stille, Nielsen transformations and applications: a survey, Groups—Korea ’94 (Pusan), de Gruyter, Berlin, 1995, pp. 69–105. MR 1476950
- Benjamin Fine, Alexei Myasnikov, Volkmar große Rebel, and Gerhard Rosenberger, A classification of conjugately separated abelian, commutative transitive, and restricted Gromov one-relator groups, Results Math. 50 (2007), no. 3-4, 183–193. MR 2343587, DOI 10.1007/s00025-007-0245-5
- D. Gaboriau, G. Levitt, and F. Paulin, Pseudogroups of isometries of $\textbf {R}$ and Rips’ theorem on free actions on $\textbf {R}$-trees, Israel J. Math. 87 (1994), no. 1-3, 403–428. MR 1286836, DOI 10.1007/BF02773004
- Benjamin Fine, Anthony M. Gaglione, Alexei Myasnikov, Gerhard Rosenberger, and Dennis Spellman, A classification of fully residually free groups of rank three or less, J. Algebra 200 (1998), no. 2, 571–605. MR 1610668, DOI 10.1006/jabr.1997.7205
- A. M. W. Glass, Partially ordered groups, Series in Algebra, vol. 7, World Scientific Publishing Co., Inc., River Edge, NJ, 1999. MR 1791008, DOI 10.1142/3811
- Vincent Guirardel, Limit groups and groups acting freely on $\Bbb R^n$-trees, Geom. Topol. 8 (2004), 1427–1470. MR 2119301, DOI 10.2140/gt.2004.8.1427
- Nancy Harrison, Real length functions in groups, Trans. Amer. Math. Soc. 174 (1972), 77–106. MR 308283, DOI 10.1090/S0002-9947-1972-0308283-0
- A. H. M. Hoare, On length functions and Nielsen methods in free groups, J. London Math. Soc. (2) 14 (1976), no. 1, 188–192. MR 422420, DOI 10.1112/jlms/s2-14.1.188
- A. H. M. Hoare, Nielsen methods in groups with a length function, Math. Scand. 48 (1981), no. 2, 153–164. MR 631332, DOI 10.7146/math.scand.a-11908
- Ilya Kapovich and Richard Weidmann, Two-generated groups acting on trees, Arch. Math. (Basel) 73 (1999), no. 3, 172–181. MR 1705011, DOI 10.1007/PL00000401
- Ilya Kapovich and Richard Weidmann, Nielsen methods and groups acting on hyperbolic spaces, Geom. Dedicata 98 (2003), 95–121. MR 1988426, DOI 10.1023/A:1024064029186
- Bilal Khan, Alexei G. Myasnikov, and Denis E. Serbin, On positive theories of groups with regular free length functions, Internat. J. Algebra Comput. 17 (2007), no. 1, 1–26. MR 2300402, DOI 10.1142/S0218196707003330
- O. Kharlampovich and A. Myasnikov, Irreducible affine varieties over a free group. II. Systems in triangular quasi-quadratic form and description of residually free groups, J. Algebra 200 (1998), no. 2, 517–570. MR 1610664, DOI 10.1006/jabr.1997.7184
- Olga Kharlampovich and Alexei Myasnikov, Implicit function theorem over free groups, J. Algebra 290 (2005), no. 1, 1–203. MR 2154989, DOI 10.1016/j.jalgebra.2005.04.001
- Olga Kharlampovich and Alexei G. Myasnikov, Effective JSJ decompositions, Groups, languages, algorithms, Contemp. Math., vol. 378, Amer. Math. Soc., Providence, RI, 2005, pp. 87–212. MR 2159316, DOI 10.1090/conm/378/07012
- Olga G. Kharlampovich, Alexei G. Myasnikov, Vladimir N. Remeslennikov, and Denis E. Serbin, Subgroups of fully residually free groups: algorithmic problems, Group theory, statistics, and cryptography, Contemp. Math., vol. 360, Amer. Math. Soc., Providence, RI, 2004, pp. 63–101. MR 2105437, DOI 10.1090/conm/360/06571
- O. Kharlampovich, A. Myasnikov, and D. Serbin, Regular actions on $\Lambda$-trees. Preprint.
- Valeriĭ M. Kopytov and Nikolaĭ Ya. Medvedev, Right-ordered groups, Siberian School of Algebra and Logic, Consultants Bureau, New York, 1996. MR 1393199
- Roger C. Lyndon, Length functions in groups, Math. Scand. 12 (1963), 209–234. MR 163947, DOI 10.7146/math.scand.a-10684
- Roger C. Lyndon and Paul E. Schupp, Combinatorial group theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 89, Springer-Verlag, Berlin-New York, 1977. MR 0577064
- G. S. Makanin, Equations in a free group, Izv. Akad. Nauk SSSR Ser. Mat. 46 (1982), no. 6, 1199–1273, 1344 (Russian). MR 682490
- Wilhelm Magnus, Abraham Karrass, and Donald Solitar, Combinatorial group theory, Second revised edition, Dover Publications, Inc., New York, 1976. Presentations of groups in terms of generators and relations. MR 0422434
- John W. Morgan and Peter B. Shalen, Valuations, trees, and degenerations of hyperbolic structures. I, Ann. of Math. (2) 120 (1984), no. 3, 401–476. MR 769158, DOI 10.2307/1971082
- A. Myasnikov, V. Remeslennikov, Length functions on free exponential groups. Proc. $N$ 26. IITPM SO RAN, Omsk, 1996, 1–34.
- Alexei G. Myasnikov, Vladimir N. Remeslennikov, and Denis E. Serbin, Regular free length functions on Lyndon’s free $\Bbb Z[t]$-group $F^{\Bbb Z[t]}$, Groups, languages, algorithms, Contemp. Math., vol. 378, Amer. Math. Soc., Providence, RI, 2005, pp. 37–77. MR 2159314, DOI 10.1090/conm/378/07010
- Alexei G. Myasnikov, Vladimir N. Remeslennikov, and Denis E. Serbin, Fully residually free groups and graphs labeled by infinite words, Internat. J. Algebra Comput. 16 (2006), no. 4, 689–737. MR 2258835, DOI 10.1142/S0218196706003141
- A. Nikolaev and D. Serbin, Finite index subgroups of fully residually free groups. Preprint 2009.
- David Promislow, Equivalence classes of length functions on groups, Proc. London Math. Soc. (3) 51 (1985), no. 3, 449–477. MR 805717, DOI 10.1112/plms/s3-51.3.449
- A. A. Razborov, Systems of equations in a free group, Izv. Akad. Nauk SSSR Ser. Mat. 48 (1984), no. 4, 779–832 (Russian). MR 755958
- Frank Rimlinger, Pregroups and Bass-Serre theory, Mem. Amer. Math. Soc. 65 (1987), no. 361, viii+73. MR 874086, DOI 10.1090/memo/0361
- Frank Rimlinger, A subgroup theorem for pregroups, Combinatorial group theory and topology (Alta, Utah, 1984) Ann. of Math. Stud., vol. 111, Princeton Univ. Press, Princeton, NJ, 1987, pp. 163–174. MR 895615
- Jean-Pierre Serre, Trees, Springer-Verlag, Berlin-New York, 1980. Translated from the French by John Stillwell. MR 607504
- John Stallings, Group theory and three-dimensional manifolds, Yale Mathematical Monographs, vol. 4, Yale University Press, New Haven, Conn.-London, 1971. A James K. Whittemore Lecture in Mathematics given at Yale University, 1969. MR 0415622
- Richard Weidmann, On the rank of amalgamated products and product knot groups, Math. Ann. 312 (1998), no. 4, 761–771. MR 1660235, DOI 10.1007/s002080050244
- Richard Weidmann, The Nielsen method for groups acting on trees, Proc. London Math. Soc. (3) 85 (2002), no. 1, 93–118. MR 1901370, DOI 10.1112/S0024611502013473
Additional Information
- Olga Kharlampovich
- Affiliation: Department of Mathematics and Statistics, Hunter College CUNY, 695 Park Avenue, New York, New York 10065
- MR Author ID: 191704
- Alexei Myasnikov
- Affiliation: Department of Mathematical Sciences, Stevens Institute of Technology, 1 Castle Point on Hudson, Hoboken, New Jersey 07030
- MR Author ID: 670299
- Vladimir Remeslennikov
- Affiliation: Department of Mathematics, Omsk State University, 55-A Prospect Mira, Omsk, Russia 644077
- Denis Serbin
- Affiliation: Department of Mathematical Sciences, Stevens Institute of Technology, 1 Castle Point on Hudson, Hoboken, New Jersey 07030
- Received by editor(s): August 9, 2009
- Received by editor(s) in revised form: March 22, 2010, and May 3, 2010
- Published electronically: January 31, 2012
- © Copyright 2012 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 364 (2012), 2847-2882
- MSC (2010): Primary 20E08, 20F65
- DOI: https://doi.org/10.1090/S0002-9947-2012-05376-1
- MathSciNet review: 2888231