Coxeter groups, quiver mutations and geometric manifolds

Abstract

We construct finite volume hyperbolic manifolds with large symmetry groups. The construction makes use of the presentations of finite Coxeter groups provided by Barot and Marsh, and involves mutations of quivers and diagrams defined in the theory of cluster algebras. We generalize our construction by assigning to every quiver or diagram of finite or affine type a CW-complex with a proper action of a finite (or affine) Coxeter group. These CW-complexes undergo mutations agreeing with mutations of quivers and diagrams. We also generalize the construction to quivers and diagrams originating from unpunctured surfaces and orbifolds.

Small‐volume hyperbolic manifolds constructed from Coxeter groups

“…The Coxeter diagram of $$\Lambda _5$$ depicted in Figure 11 is distinguished by having edges only of weight 3, and arising as a mutation of the quiver indexed by the Weyl group $$A_7$$. More concretely, in [6], Felikson and Tumarkin used techniques of quiver mutations from the theory of cluster algebras in order to construct manifolds whose symmetry groups contain a given finite Weyl group. In this way, they are able to determine all finite‐volume hyperbolic manifolds arising from quivers of finite type whose nodes are connected by at most one arrow (or quivers of finite type given by simply‐laced graphs).…”

“…In Section 4, we give additional information about the manifold $$M_*^5$$ and present our findings about the symmetry group $$I(M_*^5)$$ in Theorem 3. We also include a comparison with the 5‐manifold $$Q^5$$ constructed by Felikson and Tumarkin [6] by means of quiver mutations, and in Section 5, we complete the picture, providing constructions of small volume cusped hyperbolic $$n$$‐manifolds for $$n=3$$ and $$n=4$$.…”

Presentations of groups with even length relations

“…In particular, it is well known that the finite Coxeter groups can be classified via their Coxeter graphs and the class of finite Coxeter groups is precisely the class of finite reflection groups [3, Chapter VI, Section 4, Theorem 1;5,6]. The applications of Coxeter groups are widespread throughout algebra [3], analysis [9], applied mathematics [4], and geometry [7]. However, the many combinatorial properties of Coxeter groups make them an interesting topic of research in their own right (see [2]).…”

Presentations of Groups with Even Length Relations

“…The Coxeter graphs of the irreducible finite Coxeter systems are well-known and these graphs classify the finite Coxeter groups [3, Chapter VI, Section 4, Theorem 1]. The applications of Coxeter groups are widespread throughout algebra [3], analysis [18], applied mathematics [5] and geometry [12]. However, the many combinatorial properties of Coxeter groups make them an interesting topic of research in their own right (see [2]).…”