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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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