Small cancellation theory

Definition

Let $F(S)$ be the free group on a set $S$, with elements written as freely reduced words. Let $\abs{w}$ be the length of a word $w$. A set $R$ of cyclically reduced words in $F(S)$ is symmetrized if it is closed under taking inverses and cyclic permutations. Let $R$ be symmetrized. A piece relative to $R$ is a nonempty word $u$ such that two distinct elements $r_1 \ne r_2$ of $R$ both begin with $u$. That is, $r_1 = u v_1$ and $r_2 = u v_2$ with no cancellation.

1. Let $\lambda$ be a real number with $0 < \lambda < 1$. Then $R$ satisfies the condition $C'(\lambda)$ if $\abs{u} < \lambda \abs{r}$ whenever $r \in R$ and $r = u v$ with $u$ a piece.
2. For an integer $p \ge 2$, $R$ satisfies the condition $C(p)$ if no element of $R$ is a product of fewer than $p$ pieces.

A presentation $\langle S \mid R_0 \rangle$ is a small cancellation presentation of type $C'(\lambda)$ or $C(p)$ if the symmetrized closure of $R_0$ satisfies the condition. Small cancellation theory is the study of groups with such presentations.