Small cancellations over relatively hyperbolic groups and embedding theorems

By Denis V. Osin. 2004

181 citations recorded by the source corpus.

We generalize the small cancellation theory over ordinary hyperbolic groups to relatively hyperbolic settings. This generalization is then used to prove various embedding theorems for countable groups. For instance, we show that any countable torsion free group can be embedded into a finitely generated group with exactly two conjugacy classes. In particular, this gives the affirmative answer to the well-known question of the existence of a finitely generated group G other than ℤ/2ℤ such that all nontrivial elements of G are conjugate.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/586c15c748b37dce082b8c9cad28ee9b7e8d8ac4

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Discrete Mathematics and Combinatorics

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Normal subgroups in the Cremona group (2010, background): “Such hypotheses are well known in geometric group theory (see [48], [16] or [49] for example).”
Induced quasicocycles on groups with hyperbolically embedded subgroups (2012, background): “1] or Osin [23] for a finitely generated version of such an embedding).”
Small cancellation in acylindrically hyperbolic groups (2013, methodology): “Since the proof uses the same small cancellation conditions as [27] and [31], it follows from these papers that if G is non-virtually-cyclic and hyperbolic, then G can be chosen nonvirtually-cyclic and hyperbolic, and if G is hyperbolic relative to {Hλ}λ∈Λ, then G can be chosen hyperbolic relative t”
Transitivity degrees of countable groups and acylindrical hyperbolicity (2015, background): “Neumann (see [36]); finitely generated examples can be found in [51].”
Hyperbolically embedded subgroups and rotating families in groups acting on hyperbolic spaces (2011, result): “The proofs of the following three results are similar to proofs of their analogues for relatively hyperbolic groups (see [9, 104]).”
GROUPS ACTING ACYLINDRICALLY ON HYPERBOLIC SPACES (2017, methodology): “introduced by Babenko [13] in order to study geodesic growth of Riemannian manifolds. For more details and a survey of some recent results about conjugacy growth we refer to [41]. Based on ideas from [55], Hull and the author also obtained a complete description of functions that occur as conjugacy ”
Combinatorial algebra : syntax and semantics (2014, unknown): “The first three and the fifth results can be found in [260], the fourth is in [233], and the sixth is in [265] (for more see [260] and [282]).”
Acylindrically hyperbolic groups (2013, background): “Another direction is explored in [29], where the small cancellation theory in hyperbolic and relatively hyperbolic groups developed in [40,42] was generalized in the context of acylindrically hyperbolic groups.”
Twisted Burnside-Frobenius theory for discrete groups (2006, unknown): “471 (Ivanov group), and [34] (Osin group).”
Finitely Presented Wreath Products and Double Coset Decompositions (2005, background): “2]), and examples with exactly one nontrivial conjugacy class have recently been announced by Osin [26].”