# Cayley complex as universal covering space

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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## 3 Answers

These ideas have been developed to higher dimensions, with the intention of constructing resolutions of a group starting from a presentation; the method is to construct inductively the universal covering complex together with a contracting homotopy. The first stage is to choose a maximal tree in a Cayley graph. This has been implemented by Graham Ellis, see http://hamilton.nuigalway.ie/Hap/www/index.html.

You can also see a groupoid approach to the subgroup theorems, including the Kurosh Theorem, on subgroups of a free product, and a generalisation of Grusko's Theorem, in Philip Higgins' downloadable book Categories and Groupoids.

The relations with topology, and a groupoid, i.e. base point free, approach to covering spaces, are given in my book Topology and Groupoids. The key point in both books is the basic notion of covering morphism of groupoids, and its relation with actions. A virtue of this notion is that a covering map is modelled by a covering morphism, and so lifting of maps is studied via lifting of morphisms, and so is easier to follow and understand.

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Another application is the generalised version of the Alexander polynomial.

Let $G$ be a finitely-generated group, and take $(X,p)$ to be a CW-complex (say) with $\pi_1(X,p) = G$. Then take a homomorphism $\varphi : G \to F$, where $F$ is a free abelian group (maybe the maximal free abelian quotient of $G$, or just $\mathbb{Z}$, we can build the normal cover $\widetilde{X} \to X$ corresponding to $\ker \varphi$. Finally we associate to $G$ the homology group $H_1(\widetilde{X},\widetilde{p})$. This is a $\mathbb{Z}F$-module and an invariant of $G$.

It's fairly computable, and the usual knot-theoretic one is captured for $G$ the fundamental group of a knot complement and $F = G^\text{ab} (= \mathbb{Z})$.

Specifically for (adjectives) $3$-manifold groups, this more general theory gives you some lower bounds for an especially interesting invariant called the Thurston norm. It's basically the generalisation of "the degree of the Alexander polynomial is no larger than the knot genus".

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Application 1: Such a construction shows that a finitely-presented group acts freely and cocompactly on a simply connected simplicial 2-complex.

Using minimal tracks, Dunwoody deduced that finitely-presented groups are accessible, in the sense that they may be written as the fundamental group of a finite graph of groups whose edges are finite groups and whose vertices are groups with at most one end. As corollaries, we get Stallings' theorem or the stability of virtually free groups up to quasi-isometries (see Lectures on Geometric Group Theory, theorem 18.38, p. 473).

Application 2: In the article Topology of finite graphs, Stallings show that, if $X \to Y$ is an immersion from a finite graph to a finite bouquet of circles, then it is possible to add edges to $X$ in order to construct a covering $Z \to Y$. Such an argument can be used to prove that finitely generated free groups are LERF.

The construction was generalized by Wise (see for example A combinatorial theorem for special cube complexes) for Salvetti complexes, a kind of higher dimensional bouquet of circles; but Salvetti complexes are just the canonical CW-complexes associated to right-angled Artin groups (or RAAGs) with some additional cells of dimension $\geq 3$. Therefore, in the same way, separability properties may be proved for RAAGs, such as residual finiteness:

Let $\Gamma$ be a RAAG, $g \in \Gamma \backslash \{1\}$ and $S(\Gamma)$ the associated Salvetti complex. The universal covering $\widehat{S}$ of $S(\Gamma)$ is a CAT(0) cube complex (whose 2-skeleton is the Cayley complex of $\Gamma$). Thinking of $g \in \Gamma= \pi_1(S(\Gamma))$ as a loop in $S(\Gamma)$, let $\widehat{g}$ be one of its lifts in $\widehat{S}$, and $Y$ be the intersection of all half-spaces containing $\widehat{g}$. Noticing that $Y \subset \widehat{S}$ is a finite convex subcomplex, we have a local isometry $Y \to S(\Gamma)$. From Wise's construction, we get a finite covering $Z \to S(\Gamma)$ such that $Y$ is naturally a subcomplex of $Z$. Therefore, $\pi_1(Z) \leq \pi_1(S(\Gamma)) \simeq \Gamma$ is a finite-index subgroup not containing $g$.

More generally, the same argument proves that RAAGs are truncated subgroup separable (following the definition 5.2 in Elisabeth Green's thesis).

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