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Coxeter groups, quiver mutations and geometric manifolds. (English) Zbl 1398.20049

Summary: We construct finite volume hyperbolic manifolds with large symmetry groups. The construction makes use of the presentations of finite Coxeter groups provided by M. Barot and R. J. Marsh [Trans. Am. Math. Soc. 367, No. 3, 1945–1967 (2015; Zbl 1444.20026)], 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.

MSC:

20F55 Reflection and Coxeter groups (group-theoretic aspects)
13F60 Cluster algebras
16G20 Representations of quivers and partially ordered sets
51F15 Reflection groups, reflection geometries
57N15 Topology of the Euclidean \(n\)-space, \(n\)-manifolds (\(4 \leq n \leq \infty\)) (MSC2010)

Citations:

Zbl 1444.20026

Software:

CoxIter
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References:

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