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A small-cancellation presentation gives a hyperbolic group

Example

Consider the one-relator presentation

$G = ⟨ x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}, x_{7} ∣ x_{1} x_{2} x_{3} x_{4} x_{5} x_{6} x_{7} ⟩ .$

This is a finite $C^{'} (1 / 6)$ presentation and hence defines a hyperbolic group.

Facts & Assumptions

Given: The displayed presentation of $G$ .

[A1]

In the single relator $x_{1} x_{2} x_{3} x_{4} x_{5} x_{6} x_{7}$ , no nonempty subword occurs as an initial segment of two distinct cyclic conjugates or inverse cyclic conjugates, so the symmetrized presentation has no nontrivial pieces and therefore satisfies $C^{'} (1 / 6)$ vacuously.

[F1]

A presentation is the quotient of the free group by the normal closure of its relators (Group presentation by generators and relations).

[L1]

Every finite-rank free group is hyperbolic (Finite groups and free groups are hyperbolic).

Verification

technique · direct

1.1givenA1

By [A1], the displayed finite presentation satisfies the $C^{'} (1 / 6)$ condition.

1.2F1givenalgebra

The relator gives $x_{7} = (x_{1} x_{2} x_{3} x_{4} x_{5} x_{6})^{- 1}$ in $G$ , so the first six generators generate $G$ . The natural map $F (x_{1}, \dots, x_{6}) \to G$ is therefore onto. Conversely send $x_{i}$ to the identically named free generator for $1 \leq i \leq 6$ and send $x_{7}$ to $(x_{1} \dots x_{6})^{- 1}$ . The relator maps to $1$ , so [F1] factors this assignment through a homomorphism $G \to F (x_{1}, \dots, x_{6})$ . The two composites fix every respective generator, hence are identities. Thus $G ≅ F_{6}$ .

2.1L1step 1.1step 1.2∎

By [L1], $F_{6}$ is hyperbolic, and the explicit isomorphism in step 1.2 transfers its Cayley tree to $G$ with the corresponding generating set. Hence this finite $C^{'} (1 / 6)$ presentation defines a hyperbolic group, without importing the general linear-isoperimetric theorem.

Depends on

Used by

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Dependency tree · two levels

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Sources

Nicholas Touikan, An introduction to combinatorial and geometric group theory, Section 3.5 (standard reference, not scraped)