Subgroup properties of fully residually free groups

Kapovich, Ilya

doi:10.1090/S0002-9947-01-02840-9

Subgroup properties of fully residually free groups
HTML articles powered by AMS MathViewer

by Ilya Kapovich PDF

Trans. Amer. Math. Soc. 354 (2002), 335-362 Request permission

Abstract:

We prove that fully residually free groups have the Howson property, that is the intersection of any two finitely generated subgroups in such a group is again finitely generated. We also establish some commensurability properties for finitely generated fully residually free groups which are similar to those of free groups. Finally we prove that for a finitely generated fully residually free group the membership problem is solvable with respect to any finitely generated subgroup.

References

J. M. Alonso, T. Brady, D. Cooper, V. Ferlini, M. Lustig, M. Mihalik, M. Shapiro, and H. Short, Notes on word hyperbolic groups, Group theory from a geometrical viewpoint (Trieste, 1990) World Sci. Publ., River Edge, NJ, 1991, pp. 3–63. Edited by Short. MR 1170363
Hyman Bass, Covering theory for graphs of groups, J. Pure Appl. Algebra 89 (1993), no. 1-2, 3–47. MR 1239551, DOI 10.1016/0022-4049(93)90085-8
Benjamin Baumslag, Intersections of finitely generated subgroups in free products, J. London Math. Soc. 41 (1966), 673–679. MR 199247, DOI 10.1112/jlms/s1-41.1.673
Gilbert Baumslag and Frank Levin, On free metabelian products, Math. Z. 88 (1965), 375–379. MR 178065, DOI 10.1007/BF01112221
Gilbert Baumslag, Some aspects of groups with unique roots, Acta Math. 104 (1960), 217–303. MR 122859, DOI 10.1007/BF02546390
V. N. Bezverkhniĭ, Solvability of the inclusion problem in a class of HNN-groups, Algorithmic problems of the theory of groups and semigroups, Tul′sk. Gos. Ped. Inst., Tula, 1981, pp. 20–62 (Russian). MR 669218
V. N. Bezverkhniĭ, Solution of the occurrence problem in some classes of groups with one defining relation, Algorithmic problems in the theory of groups and semigroups (Russian), Tul′sk. Gos. Ped. Inst., Tula, 1986, pp. 3–21, 126 (Russian). MR 932268
V. N. Bezverkhniĭ, Solution of the occurrence problem in a class of groups. I, Problems in group theory and homological algebra (Russian), Matematika, Yaroslav. Gos. Univ., Yaroslavl′1990, pp. 40–53 (Russian). MR 1169963
V. N. Bezverkhniĭ, Solution of the occurrence problem in a certain class of groups. II, Problems in group theory and homological algebra (Russian), Yaroslav. Gos. Univ., Yaroslavl′1991, pp. 122–142 (Russian). MR 1217539
R. G. Burns, On the finitely generated subgroups of an amalgamated product of two groups, Trans. Amer. Math. Soc. 169 (1972), 293–306. MR 372043, DOI 10.1090/S0002-9947-1972-0372043-5
R. G. Burns, Finitely generated subgroups of $\textbf {HNN}$ groups, Canadian J. Math. 25 (1973), 1103–1112. MR 330305, DOI 10.4153/CJM-1973-117-7
M. Bestvina and M. Feighn, A combination theorem for negatively curved groups, J. Differential Geom. 35 (1992), no. 1, 85–101. MR 1152226, DOI 10.4310/jdg/1214447806
G. Baumslag, S. M. Gersten, M. Shapiro, and H. Short, Automatic groups and amalgams, J. Pure Appl. Algebra 76 (1991), no. 3, 229–316. MR 1147304, DOI 10.1016/0022-4049(91)90139-S
D. E. Cohen, Subgroups of HNN groups, J. Austral. Math. Soc. 17 (1974), 394–405. Collection of articles dedicated to the memory of Hanna Neumann, VIII. MR 0450408, DOI 10.1017/S1446788700018036
Daniel E. Cohen, Finitely generated subgroups of amalgamated free products and HNN groups, J. Austral. Math. Soc. Ser. A 22 (1976), no. 3, 274–281. MR 439940, DOI 10.1017/s1446788700014737
Daniel E. Cohen, Combinatorial group theory: a topological approach, London Mathematical Society Student Texts, vol. 14, Cambridge University Press, Cambridge, 1989. MR 1020297, DOI 10.1017/CBO9780511565878
M. Coornaert, T. Delzant, and A. Papadopoulos, Géométrie et théorie des groupes, Lecture Notes in Mathematics, vol. 1441, Springer-Verlag, Berlin, 1990 (French). Les groupes hyperboliques de Gromov. [Gromov hyperbolic groups]; With an English summary. MR 1075994, DOI 10.1007/BFb0084913
David B. A. Epstein, James W. Cannon, Derek F. Holt, Silvio V. F. Levy, Michael S. Paterson, and William P. Thurston, Word processing in groups, Jones and Bartlett Publishers, Boston, MA, 1992. MR 1161694, DOI 10.1201/9781439865699
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. Gaglione, A. G. Myasnikov, V. N. Remeslennikov, and D. Spellman, Formal power series representations of free exponential groups, Comm. Algebra 25 (1997), no. 2, 631–648. MR 1428804, DOI 10.1080/00927879708825880
Anthony M. Gaglione and Dennis Spellman, Even more model theory of free groups, Infinite groups and group rings (Tuscaloosa, AL, 1992) Ser. Algebra, vol. 1, World Sci. Publ., River Edge, NJ, 1993, pp. 37–40. MR 1377955
S. M. Gersten and H. B. Short, Rational subgroups of biautomatic groups, Ann. of Math. (2) 134 (1991), no. 1, 125–158. MR 1114609, DOI 10.2307/2944334
Nelson Dunford, A mean ergodic theorem, Duke Math. J. 5 (1939), 635–646. MR 98
Rita Gitik, On quasiconvex subgroups of negatively curved groups, J. Pure Appl. Algebra 119 (1997), no. 2, 155–169. MR 1453217, DOI 10.1016/S0022-4049(96)00020-5
M. Gromov, Hyperbolic groups, Essays in group theory, Math. Sci. Res. Inst. Publ., vol. 8, Springer, New York, 1987, pp. 75–263. MR 919829, DOI 10.1007/978-1-4613-9586-7_{3}
É. Ghys and P. de la Harpe (eds.), Sur les groupes hyperboliques d’après Mikhael Gromov, Progress in Mathematics, vol. 83, Birkhäuser Boston, Inc., Boston, MA, 1990 (French). Papers from the Swiss Seminar on Hyperbolic Groups held in Bern, 1988. MR 1086648, DOI 10.1007/978-1-4684-9167-8
D. Gildenhuys, O. Kharlampovich, and A. Myasnikov, CSA-groups and separated free constructions, Bull. Austral. Math. Soc. 52 (1995), no. 1, 63–84. MR 1344261, DOI 10.1017/S0004972700014453
Rita Gitik, Mahan Mitra, Eliyahu Rips, and Michah Sageev, Widths of subgroups, Trans. Amer. Math. Soc. 350 (1998), no. 1, 321–329. MR 1389776, DOI 10.1090/S0002-9947-98-01792-9
S. G. Ivanov, The membership problem for a free product of groups with an amalgamated subgroup, Sibirsk. Mat. Ž. 16 (1975), no. 6, 1155–1171, 1369 (Russian). MR 0419621
Ilya Kapovich, Detecting quasiconvexity: algorithmic aspects, Geometric and computational perspectives on infinite groups (Minneapolis, MN and New Brunswick, NJ, 1994) DIMACS Ser. Discrete Math. Theoret. Comput. Sci., vol. 25, Amer. Math. Soc., Providence, RI, 1996, pp. 91–99. MR 1364182, DOI 10.1090/dimacs/025/07
Ilya Kapovich, Amalgamated products and the Howson property, Canad. Math. Bull. 40 (1997), no. 3, 330–340. MR 1464841, DOI 10.4153/CMB-1997-039-3
Ilya Kapovich, Quasiconvexity and amalgams, Internat. J. Algebra Comput. 7 (1997), no. 6, 771–811. MR 1482968, DOI 10.1142/S0218196797000344
Ilya Kapovich, A non-quasiconvexity embedding theorem for hyperbolic groups, Math. Proc. Cambridge Philos. Soc. 127 (1999), no. 3, 461–486. MR 1713122, DOI 10.1017/S0305004199003862
O. Kharlampovich and A. Myasnikov, Equations in a free $\textbf {Q}$-group, Trans. Amer. Math. Soc. 350 (1998), no. 3, 947–974. MR 1389782, DOI 10.1090/S0002-9947-98-01798-X
O. Kharlampovich and A. Myasnikov, Hyperbolic groups and free constructions, Trans. Amer. Math. Soc. 350 (1998), no. 2, 571–613. MR 1390041, DOI 10.1090/S0002-9947-98-01773-5
Olga Kharlampovich and Alexei Myasnikov, Tarski’s problem about the elementary theory of free groups has a positive solution, Electron. Res. Announc. Amer. Math. Soc. 4 (1998), 101–108. MR 1662319, DOI 10.1090/S1079-6762-98-00047-X
O. Kharlampovich and A. Myasnikov, Irreducible affine varieties over a free group. I. Irreducibility of quadratic equations and Nullstellensatz, J. Algebra 200 (1998), no. 2, 472–516. MR 1610660, DOI 10.1006/jabr.1997.7183
O. Kharlampovich and A. Myasnikov, Irreducible affine varieties over a free group. I. Irreducibility of quadratic equations and Nullstellensatz, J. Algebra 200 (1998), no. 2, 472–516. MR 1610660, DOI 10.1006/jabr.1997.7183
O. Kharlampovich and A. Myasnikov, Description of fully residually free groups and irreducible affine varieties over a free group, Summer School in Group Theory in Banff, 1996, CRM Proc. Lecture Notes, vol. 17, Amer. Math. Soc., Providence, RI, 1999, pp. 71–80. MR 1653685, DOI 10.1090/crmp/017/04
O. Kharlampovich and A. Myasnikov, Implicit function theorem over free groups, preprint, McGill University, 1999; http://www.math.mcgill.ca/olga/publications.html
O. Kharlampovich and A. Myasnikov, Equations over fully residually free groups, preprint, McGill University, 1999; http://www.math.mcgill.ca/olga/publications.html
O. Kharlampovich and A. Myasnikov, Elementary theory of free nonabelian groups, preprint, McGill University, 1999; http://www.math.mcgill.ca/olga/publications.html
Ilya Kapovich and Hamish Short, Greenberg’s theorem for quasiconvex subgroups of word hyperbolic groups, Canad. J. Math. 48 (1996), no. 6, 1224–1244. MR 1426902, DOI 10.4153/CJM-1996-065-6
A. Karrass and D. Solitar, The subgroups of a free product of two groups with an amalgamated subgroup, Trans. Amer. Math. Soc. 150 (1970), 227–255
Ilya Kapovich and Richard Weidmann, On the structure of two-generated hyperbolic groups, Math. Z. 231 (1999), no. 4, 783–801. MR 1709496, DOI 10.1007/PL00004753
I. Kapovich and D. Wise, The equivalence of some residual properties of word-hyperbolic groups, J. Algebra 223 (2000), no. 2, 562–583
Roger C. Lyndon, Groups with parametric exponents, Trans. Amer. Math. Soc. 96 (1960), 518–533. MR 151502, DOI 10.1090/S0002-9947-1960-0151502-6
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
K. A. Mihaĭlova, The occurrence problem for free products of groups, Mat. Sb. (N.S.) 75 (117) (1968), 199–210 (Russian). MR 0222179
Ch. F. Miller III, Decision problems in algebraic classes of groups (a survey), Word problems: decision problems and the Burnside problem in group theory (Conf., Univ. California, Irvine, Calif. 1969; dedicated to Hanna Neumann), Studies in Logic and the Foundations of Math., Vol. 71, North-Holland, Amsterdam, 1973, pp. 507–523. MR 0399270
A.Myasnikov, Subgroups of free exponential groups, City College of CUNY, preprint, 1999
A. Karrass, W. Magnus, and D. Solitar, Elements of finite order in groups with a single defining relation, Comm. Pure Appl. Math. 13 (1960), 57–66. MR 124384, DOI 10.1002/cpa.3160130107
A. G. Myasnikov and V. N. Remeslennikov, Degree groups. I. Foundations of the theory and tensor completions, Sibirsk. Mat. Zh. 35 (1994), no. 5, 1106–1118, iii (Russian, with Russian summary); English transl., Siberian Math. J. 35 (1994), no. 5, 986–996. MR 1308240, DOI 10.1007/BF02104576
A. G. Myasnikov and V. N. Remeslennikov, Exponential groups. II. Extensions of centralizers and tensor completion of CSA-groups, Internat. J. Algebra Comput. 6 (1996), no. 6, 687–711. MR 1421886, DOI 10.1142/S0218196796000398
Michael L. Mihalik and Williams Towle, Quasiconvex subgroups of negatively curved groups, J. Pure Appl. Algebra 95 (1994), no. 3, 297–301. MR 1295962, DOI 10.1016/0022-4049(94)90063-9
J. McCammond and D. Wise, Coherence, local quasiconvexity and the perimeter of 2-complexes, preprint, 1998
A. Yu. Ol′shanskiĭ, On residualing homomorphisms and $G$-subgroups of hyperbolic groups, Internat. J. Algebra Comput. 3 (1993), no. 4, 365–409. MR 1250244, DOI 10.1142/S0218196793000251
V. N. Remeslennikov, $\exists$-free groups and groups with length function, Second International Conference on Algebra (Barnaul, 1991) Contemp. Math., vol. 184, Amer. Math. Soc., Providence, RI, 1995, pp. 369–376. MR 1332302, DOI 10.1090/conm/184/02131
Z. Sela, Diophantine geometry over groups I: Makanin-Razborov diagrams, Hebrew University, preprint; http://www.ma.huji.ac.il/~zlil/
Z. Sela, Diophantine geometry over groups II: Completions, closures and formal solutions, Hebrew University, preprint; http://www.ma.huji.ac.il/~zlil/
Z. Sela, Diophantine geometry over groups III: Rigid and solid solutions, Hebrew University, preprint; http://www.ma.huji.ac.il/~zlil/
Z. Sela, Diophantine geometry over groups IV: An iterative procedure for validation of a sentence, Hebrew University, preprint, http://www.ma.huji.ac.il/~zlil/
Z. Sela, Diophantine geometry over groups V: Quantifier elimination, Hebrew University, preprint; http://www.ma.huji.ac.il/~zlil/
Z. Sela, Diophantine geometry over groups VI: The elementary theory of a free group, Hebrew University, preprint; http://www.ma.huji.ac.il/~zlil/
Jean-Pierre Serre, Trees, Springer-Verlag, Berlin-New York, 1980. Translated from the French by John Stillwell. MR 607504, DOI 10.1007/978-3-642-61856-7
Hamish Short, Quasiconvexity and a theorem of Howson’s, Group theory from a geometrical viewpoint (Trieste, 1990) World Sci. Publ., River Edge, NJ, 1991, pp. 168–176. MR 1170365
E. Swenson, Quasi-convex groups of isometries of negatively curved spaces, Topol. Appl. 110 (2001), no. 1, 119–129