Geodesic
Low dimensional homotopy for monoids II: groups
Glasgow mathematical journal, Tome 41 (1999) no. 1, pp. 1-11

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Consider a group presentation: $$\hat{[Pscr]}\tfrm{=<\tfbf{x};}\tfbf{r}\tfrm{>}$$. 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. 
PRIDE, STEPHEN J. Low dimensional homotopy for monoids II: groups. Glasgow mathematical journal, Tome 41 (1999) no. 1, pp. 1-11. doi: 10.1017/S0017089599970179
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