Geodesic
Pregroups and the big powers condition
Algebra i logika, Tome 48 (2009) no. 3, pp. 342-377.

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We study groups having the big powers property BP. It is proved that if a pregroup satisfies some natural axioms, then its universal group has this property. In particular, fundamental groups of some graphs of groups have the big powers property if BP holds for edge and vertex subgroups and a number of natural conditions are satisfied. The results obtained are applied to Lyndon's completions $U(P)^{\mathbb Z[t]}$ of the universal group $U(P)$ with $P$ satisfying the conditions mentioned.
Mots-clés : group
Keywords: pregroup, universal group, big powers property. 
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A. V. Kvaschuk; A. G. Myasnikov; D. E. Serbin. Pregroups and the big powers condition. Algebra i logika, Tome 48 (2009) no. 3, pp. 342-377. http://geodesic.mathdoc.fr/item/AL_2009_48_3_a2/

[1] A. G. Myasnikov, V. N. Remeslennikov, Length functions on free exponential groups, Preprint No 26, IITPM SB RAS, Omsk, 1996, 34 pp. | MR

[2] A. G. Myasnikov, V. N. Remeslennikov, Big powers and free-like groups, Preprint, IITPM SB RAS, Omsk, 1999

[3] A. Yu. Ol'shanskij, “On residualing homomorphisms and $G$-subgroups of hyperbolic groups”, Int. J. Algebra Comput., 3:4 (1993), 365–409 | DOI | MR

[4] G. Baumslag, A. Myasnikov, V. Remeslennikov, “Discriminating completions of hyperbolic groups”, Geom. Dedicata, 92 (2002), 115–143 | DOI | MR | Zbl

[5] R. Lyndon, “Equations in free groups”, Trans. Am. Math. Soc., 96 (1960), 445–457 | DOI | MR | Zbl

[6] K. I. Appel, “One-variable equations in free groups”, Proc. Am. Math. Soc., 19 (1968), 912–918 | DOI | MR | Zbl

[7] G. S. Makanin, “Uravneniya v svobodnykh gruppakh”, Izv. AN SSSR. Ser. matem., 46:6 (1982), 1199–1273 | MR | Zbl

[8] G. Baumslag, “On generalised free products”, Math. Zeitschr., 78:5 (1962), 423–438 | DOI | MR | Zbl

[9] A. Myasnikov, V. Remeslennikov, D. Serbin, “Regular free length functions on Lyndon's free $\mathbb Z[t]$-group $F^{\mathbb Z[t]}$”, Groups, languages, algorithms, Proc. the AMS-ASL joint special session on interactions between logic, group theory, and computer science (Baltimore, MD, USA, Jan. 16–19, 2003), Contemp. Math., 378, ed. A. V. Borovik, Am. Math. Soc., Providence, RI, 2005, 37–77 | MR | Zbl

[10] O. Kharlampovich, A. Myasnikov, “Implicit function theorems over free groups”, J. Algebra, 290:1 (2005), 1–203 | DOI | MR | Zbl

[11] A. G. Myasnikov, V. N. Remeslennikov, “Exponential groups. II: Extensions of centralizers and tensor completion of CSA-groups”, Int. J. Algebra Comput., 6:6 (1996), 687–711 | DOI | MR | Zbl

[12] O. Kharlampovich, A. Myasnikov, “Irreducible affine varieties over a free group. I: Irreducibility of quadratic equations and Nullstellensatz”, J. Algebra, 200:2 (1998), 472–516 | DOI | MR | Zbl

[13] O. Kharlampovich, A. Myasnikov, “Irreducible affine varieties over a free group. II: Systems in triangular quasi-quadratic form and description of residually free groups”, J. Algebra, 200:2 (1998), 517–570 | DOI | MR | Zbl

[14] O. Kharlampovich, A. Myasnikov, Limits of relatively hyperbolic groups and Lyndon's completions, Preprint, 2009 | Zbl

[15] M. Gromov, “Hyperbolic groups”, Essays in group theory, Publ. Math. Sci. Res. Inst., 8, ed. S. M. Gersten, 1987, 75–263 | MR | Zbl

[16] J. M. Alonso, T. Brady, D. Cooper, V. Ferlini, M. Lustig, M. Mihalik, M. Shapiro, H. Short, “Notes on word hyperbolic groups”, Group theory from a geometrical viewpoint, Proc. of a workshop, held at the Inter. Centre for Theor. Phys. (Trieste, Italy, 26 March to 6 April 1990), eds. E. Ghys et al., World Scientific, Singapore, 1991, 3–63 | MR | Zbl

[17] A. Yu. Olshanskii, “Periodicheskie faktor-gruppy giperbolicheskikh grupp”, Matem. sb., 182:4 (1991), 543–567 | MR | Zbl

[18] J. R. Stallings, “Groups of cohomological dimension one”, Appl. categorical Algebra, Proc. Sympos. Pure Math., 17, 1970, 124–128 | MR | Zbl

[19] J. R. Stallings, Group theory and three dimensional manifolds, Yale Math. Monogr., 4, Yale Univ. Press, New Haven–London, 1971 | MR | Zbl

[20] A. H. M. Hoare, “Pregroups and length functions”, Math. Proc. Camb. Philos. Soc., 104:1 (1988), 21–30 | DOI | MR | Zbl

[21] I. M. Chiswell, “Length functions and pregroups”, Proc. Edinb. Math. Soc. II Ser., 30 (1987), 57–67 | DOI | MR | Zbl

[22] F. H. Nesayef, Groups generated by elements of length zero and one, Ph. D. Thesis, Univ. Birmingham, 1983

[23] D. Promislow, “Equivalence classes of length functions on groups”, Proc. Lond. Math. Soc. III Ser., 51:3 (1985), 449–477 | DOI | MR | Zbl

[24] D. Gildenhuys, O. Kharlampovich, A. Myasnikov, “CSA-groups and separated free constructions”, Bull. Aust. Math. Soc., 52:1 (1995), 63–84 | DOI | MR | Zbl

[25] R. Lyndon, “Groups with parametric exponents”, Trans. Am. Math. Soc., 96 (1960), 518–533 | DOI | MR | Zbl

[26] G. Baumslag, A. Myasnikov, V. Remeslennikov, “Algebraic geometry over groups. I: Algebraic sets and ideal theory”, J. Algebra, 219:1 (1999), 16–79 | DOI | MR | Zbl