Independent Artin fundamental-group identification

Let $E$ be a finite-dimensional real vector space and $\bar{C}$ a closed convex polyhedral cone with nonempty interior $C$. A wall is the hyperplane spanned by a codimension-one face of $\bar{C}$. For each wall choose a linear reflection fixing it, meaning a linear transformation of order two whose fixed space is that wall. Let $S$ be the set of these reflections and let $W$ be the subgroup of $\mathrm{G}\mathrm{L}(E)$ generated by $S$. Assume $wC\cap C=\mathrm{\varnothing }$ for every nonidentity $w\in W$; this is a Vinberg system.

Put $T=\mathrm{int}({\bigcup }_{w\in W}w\bar{C})$ and let $\mathcal{A}$ be the set of fixed hyperplanes of all reflections in $W$. Then $T$ is a nonempty open convex cone, $\mathcal{A}$ is a locally finite arrangement in $T$, and $W$ preserves both. Give $M=(T\times T)\setminus {\bigcup }_{H\in \mathcal{A}}(H\times H)$ the subspace topology and let $W$ act diagonally. This action is free and properly discontinuous and is a covering-space action. The space $M$ is path connected, and $M\to M\mathrm{/}W$ is a regular covering with deck group $W$.

The pair $(W,S)$ is a Coxeter system. For distinct $s,t\in S$, let $m(s,t)$ be the order of $st$, possibly infinite. Let $A$ be the group generated by ${\sigma }_s$ for $s\in S$, with the relation equating the two alternating words of length $m(s,t)$ beginning respectively with ${\sigma }_s$ and ${\sigma }_t$ whenever $m(s,t)$ is finite. Let $\theta :A\to W$ send ${\sigma }_s$ to $s$, and put $P=\ker \theta$. For a basepoint $z\in M$ and its image $\bar{z}\in M\mathrm{/}W$, the covering sequence $1\to {\pi }_1(M,z)\to {\pi }_1(M\mathrm{/}W,\bar{z})\to W\to 1$ is isomorphic, as a sequence over $W$, to $1\to P\to A\stackrel{\theta }{\to }W\to 1$. Here ${\pi }_1$ is the group of based loop homotopy classes, and the map to $W$ records the endpoint translate of a lifted loop.

Complexified complements in an open convex cone

Artin monoid and Artin group presentations, and the canonical monoid-to-group map

Fundamental groupoid of a space

The quotient topology of a surjection, quotient maps, saturated sets, and the quotient of a space by an equivalence relation with its canonical projection

Covering maps, evenly covered neighbourhoods, fibres, sheets, and trivial coverings

Regular coverings

Deck transformations and the deck-transformation group of a covering

Lifts of maps, paths, and homotopies through a covering map

Paris states Vinberg's results on the cone and arrangement, states that the diagonal action on the complement is free and properly discontinuous, and gives Van der Lek's identification of the regular-covering sequence with $1\to P\to A\stackrel{\theta }{\to }W\to 1$ under the Vinberg-system hypotheses above. The cited complement is connected. The imported identification does not assert that the complement or its quotient is aspherical.

The source statements are used as external inputs; their proofs are not reproduced or independently reviewed here.