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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
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