Low dimensional homotopy for monoids II: groups
Stephen J. Pride · Glasgow Mathematical Journal · 1999
Consider a group presentation: $$\hat{[Pscr ]}\tfrm{=}$$. Here x is a set and r is a set of non-empty, cyclically reduced words on the alphabet x ∪ x−1 (where x−1 is a set in one-to-one correspondence x[harr ]x−1 with x). We assume throughout that $\hat{[Pscr ]}$ is finite. Let $\hat{F}$ be the free group on x (thus $\hat{F}$ consists of free equivalence classes [W] of word on x∪x−1), and let N be the normal closure of {[R] : R∈r} in $\hat{F}$. Then the group G=G($\hat{[Pscr ]}$) defined by $\hat{[Pscr ]}$ is $\hat{F}\tfrm{/}N$. We will write W1 =GW2 if [W1]N=[W2]N.