[Paper Review] Small cancellations over relatively hyperbolic groups and embedding theorems

Denis Osin|arXiv (Cornell University)|Nov 2, 2004

Geometric and Algebraic Topology7 references17 citations

TL;DR

This paper generalizes small cancellation theory to relatively hyperbolic groups, enabling new embedding theorems for countable groups. It proves that any countable torsion-free group embeds into a 2-generated group with exactly two conjugacy classes—resolving a long-standing open problem about the existence of finitely generated groups (other than ℤ/2ℤ) with precisely two conjugacy classes.

ABSTRACT

We generalize the small cancellation theory over hyperbolic groups developed by Olshanskii to the case of relatively hyperbolic groups. This allows us to construct infinite finitely generated groups with exactly $n$ conjugacy classes for every $n\ge 2$. In particular, we give the affirmative answer to the well--known question of the existence of a finitely generated group $G$ other than $\mathbb Z/2\mathbb Z$ such that all nontrivial elements of $G$ are conjugate.

Motivation & Objective

• To extend small cancellation theory from ordinary hyperbolic groups to relatively hyperbolic settings.
• To address the open problem of whether there exists a finitely generated group (other than ℤ/2ℤ) with exactly two conjugacy classes.
• To construct finitely generated groups with prescribed conjugacy class structures, including exactly n conjugacy classes for any n ≥ 2.
• To show that any countable group embeds into a 2-generated verbally complete group, preserving torsion-freeness when applicable.
• To establish that such groups cannot arise as limits of hyperbolic groups, highlighting structural distinctions.

Proposed method

• Generalizes Olshanskii’s small cancellation over hyperbolic groups to relatively hyperbolic groups using techniques from prior work by the author.
• Applies the generalized small cancellation to construct group extensions via HNN-like constructions and controlled diagrammatic analysis.
• Uses contiguity subdiagrams and metric estimates (e.g., (Π,Γ,i,*)-measures) to control overlaps and ensure small cancellation conditions.
• Employs inductive arguments on the number of R-cells in diagrams to derive contradictions when multiple edges or loops exist in the contiguity graph.
• Applies quasi-geodesic properties and length bounds (e.g., l(u) ≤ λ⁻¹(3ε + c)) to bound contiguity measures and enforce small cancellation.
• Uses the structure of the contiguity graph Φ_M and its edge/vertex properties to enforce minimality and derive contradictions under assumptions of multiple edges or loops.

Experimental results

Research questions

• RQ1 Can small cancellation theory be extended from hyperbolic groups to relatively hyperbolic groups?
• RQ2 Does there exist a finitely generated group with exactly two conjugacy classes other than ℤ/2ℤ?
• RQ3 Can any countable group be embedded into a 2-generated group where elements of the same order are conjugate?
• RQ4 Can any countable group be embedded into a finitely generated verbally complete group?
• RQ5 Are finitely generated groups with exactly n conjugacy classes (n ≥ 2) constructible as limits of hyperbolic groups?

Key findings

• Any countable group embeds into a 2-generated group where elements of the same order are conjugate and the set of finite orders is preserved.
• Any countable torsion-free group embeds into a 2-generated torsion-free group with exactly two conjugacy classes.
• There exists an uncountable family of pairwise non-isomorphic 2-generated torsion-free groups with exactly two conjugacy classes.
• For any n ≥ 2, there exists an uncountable family of pairwise non-isomorphic finitely generated groups with exactly n conjugacy classes.
• No finitely generated group with exactly two conjugacy classes (other than ℤ/2ℤ) can be constructed as a limit of hyperbolic groups.
• Any countable group embeds into a 2-generated verbally complete group, and if the original group is torsion-free, the target can be chosen torsion-free as well.