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In Combinatorial Group Theory, Lyndon and Schupp construct a complex $K(X;R)$ from a presentation of group $G=(X;R)$, such that $G \simeq \pi_1(K,v)$ (proposition 2.3, p.117). Moreover, the Cayley complex of $G$ is the universal covering space of $K(X;R)$ (proposition 4.3, p. 124).

The only application I found in the book is the subgroup theorem of Schreier and Nielsen (every subgroup of a free group is free).

Are there any other interesting applications?

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