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Small cancellations over relatively hyperbolic groups and embedding theorems. (English) Zbl 1203.20031
Small cancellation theory over hyperbolic groups is generalized to relatively hyperbolic groups. This is used to prove:
Theorem 1.1. Any countable group $$G$$ can embed into a 2-generated group $$C$$ such that any two elements of the same order are conjugate in $$C$$ and $$\pi(G)=\pi(C)$$.
Here $$\pi(G)$$ is the set of all finite orders of elements of a group $$G$$.
Corollary 1.2. Any countable torsion-free group can embed into a torsion-free 2-generated group with exactly 2 conjugacy classes.
The existence of a finitely generated group with exactly 2 conjugacy classes other than $$\mathbb{Z}/2\mathbb{Z}$$ was open before.
Corollary 1.3. There exists an uncountable set of pairwise nonisomorphic torsion-free 2-generated groups with exactly 2 conjugacy classes.
Corollary 1.4. For any natural number $$n\geq 2$$ there is an uncountable set of pairwise nonisomorphic finitely generated groups with exactly $$n$$ conjugacy classes.
A ‘verbally complete group’ is a group $$W$$ such that for every nontrivial freely reduced word $$w(x_i)$$ in the alphabet $$x_1^{\pm 1},x_2^{\pm 1},\dots$$ and every $$v\in W$$, the equation $$w(x_i)=v$$ has a solution in $$W$$.
Theorem 1.5. Any countable group $$H$$ can be embedded into a 2-generated verbally complete group $$W$$. If $$H$$ is torsion-free then $$W$$ can be chosen to be torsion-free.
Corollary 1.6. There exists an uncountable set of pairwise nonisomorphic 2-generated verbally complete groups.

MSC:
 20F05 Generators, relations, and presentations of groups 20E07 Subgroup theorems; subgroup growth 20F67 Hyperbolic groups and nonpositively curved groups 20E45 Conjugacy classes for groups 20F06 Cancellation theory of groups; application of van Kampen diagrams 20F65 Geometric group theory 20F70 Algebraic geometry over groups; equations over groups
MathOverflow Questions:
Commutator problem vs conjugacy/word problem
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References:
  G. Arzhantseva, M. R. Bridson, T. Januszkiewicz, I. J. Leary, A. Minasyan, and J. Świcatkowski, ”Infinite groups with fixed point properties,” Geom. Topol., vol. 13, iss. 3, pp. 1229-1263, 2009. · Zbl 1197.20034  G. Arzhantseva, A. Minasyan, and D. Osin, ”The SQ-universality and residual properties of relatively hyperbolic groups,” J. Algebra, vol. 315, iss. 1, pp. 165-177, 2007. · Zbl 1132.20022  G. Baumslag, A. G. Myasnikov, and V. Shpilrain, ”Open problems in combinatorial group theory,” in Combinatorial and Geometric Group Theory (New York, 2000/Hoboken, NJ, 2001), Second ed., Providence, RI: Amer. Math. Soc., 2002, pp. 1-38. · Zbl 1065.20042  I. Belegradek and D. Osin, ”Rips construction and Kazhdan property (T),” Groups Geom. Dyn., vol. 2, iss. 1, pp. 1-12, 2008. · Zbl 1152.20039  B. H. Bowditch, ”Relatively hyperbolic groups,” , preprint , 1999. · Zbl 1259.20052  M. R. Bridson and A. Haefliger, Metric Spaces of Non-Positive Curvature, New York: Springer-Verlag, 1999. · Zbl 0988.53001  F. Dahmani, ”Combination of convergence groups,” Geom. Topol., vol. 7, pp. 933-963, 2003. · Zbl 1037.20042  B. Farb, ”Relatively hyperbolic groups,” Geom. Funct. Anal., vol. 8, iss. 5, pp. 810-840, 1998. · Zbl 0985.20027  S. M. Gersten and H. B. Short, ”Rational subgroups of biautomatic groups,” Ann. of Math., vol. 134, iss. 1, pp. 125-158, 1991. · Zbl 0744.20035  Sur les Groupes Hyperboliques d’après Mikhael Gromov, Ghys, E. and de la Harpe, P., Eds., Boston, MA: Birkhäuser, 1990, vol. 83.  M. Gromov, ”Hyperbolic groups,” in Essays in Group Theory, New York: Springer-Verlag, 1987, vol. 8, pp. 75-263. · Zbl 0634.20015  V. S. Guba, ”A finitely generated complete group,” Izv. Akad. Nauk SSSR Ser. Mat., vol. 50, iss. 5, pp. 883-924, 1986. · Zbl 0631.20025  G. Higman, B. H. Neumann, and H. Neumann, ”Embedding theorems for groups,” J. London Math. Soc., vol. 24, pp. 247-254, 1949. · Zbl 0034.30101  S. V. Ivanov, ”On some finiteness conditions in semigroup and group theory,” Semigroup Forum, vol. 48, iss. 1, pp. 28-36, 1994. · Zbl 0808.20046  S. V. Ivanov and A. Y. Ol$$'$$shanskii, ”Some applications of graded diagrams in combinatorial group theory,” in Groups-St. Andrews 1989, Vol. 2, Cambridge: Cambridge Univ. Press, 1991, pp. 258-308.  The Kourovka Notebook. Unsolved Problems in Group Theory, fifteenth augmented ed., Mazurov, V. D. and Khukhro, E. I., Eds., Institut Matematiki im. S. L. Soboleva, Novosibirsk: Rossiĭskaya Akademiya Nauk Sibirskoe Otdelenie, 2002.  R. C. Lyndon and P. E. Schupp, Combinatorial Group Theory, New York: Springer-Verlag, 1977. · Zbl 0368.20023  K. V. Mikhajlovskii and Y. A. Ol$$'$$shanskii, ”Some constructions relating to hyperbolic groups,” in Geometry and Cohomology in Group Theory, Cambridge: Cambridge Univ. Press, 1998, pp. 263-290. · Zbl 0910.20025  A. Minasyan, ”Groups with finitely many conjugacy classes and their automorphisms,” Comment. Math. Helv., vol. 84, iss. 2, pp. 259-296, 2009. · Zbl 1180.20033  A. Minasyan and D. Osin, ”Normal automorphisms of hyperbolic groups,” , preprint , 2008. · Zbl 1227.20041  . A. Ol$$'$$shanskiui, Geometry of Defining Relations in Groups, Dordrecht: Kluwer Academic Publishers Group, 1991. · Zbl 0830.20053  . A. Ol$$'$$shanskiui, ”Periodic quotient groups of hyperbolic groups,” Mat. Sb., vol. 182, iss. 4, pp. 543-567, 1991.  . A. Ol$$'$$shanskiui, ”On residualing homomorphisms and $$G$$-subgroups of hyperbolic groups,” Internat. J. Algebra Comput., vol. 3, iss. 4, pp. 365-409, 1993. · Zbl 0830.20053  . A. Ol$$'$$shanskiui, ”On the Bass-Lubotzky question about quotients of hyperbolic groups,” J. Algebra, vol. 226, iss. 2, pp. 807-817, 2000. · Zbl 0957.20026  D. V. Osin, ”Relatively hyperbolic groups: intrinsic geometry, algebraic properties, and algorithmic problems,” Mem. Amer. Math. Soc., vol. 179, iss. 843, p. vi, 2006. · Zbl 1093.20025  D. V. Osin, ”Elementary subgroups of relatively hyperbolic groups and bounded generation,” Internat. J. Algebra Comput., vol. 16, iss. 1, pp. 99-118, 2006. · Zbl 1100.20033  D. V. Osin, ”Relative Dehn functions of amalgamated products and HNN-extensions,” in Topological and Asymptotic Aspects of Group Theory, Providence, RI: Amer. Math. Soc., 2006, pp. 209-220. · Zbl 1111.20036  D. V. Osin, Factorizable groups and finiteness conditions.
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