Coxeter groups, quiver mutations and geometric manifolds

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.


Introduction
In [BM] Barot and Marsh provided presentations of finite Weyl groups in terms of arbitrary quiver or diagram of finite type.In brief, the construction works as follows: one needs to consider the underlying unoriented labeled graph of a quiver (or diagram) as a Coxeter diagram of a (usually infinite) Coxeter group, and then impose some additional relations on this group that can be read off from the quiver/diagram.It is proved in [BM] that the resulting group depends on the mutation class of a quiver/diagram only, see Section 3 for the details.
The goal of this paper is to use these presentations to construct hyperbolic manifolds, in particular ones of small volume.The most common way to construct hyperbolic manifolds is by considering finite index torsion-free subgroups of cofinite hyperbolic reflection groups (see e.g.[CM, E, ERTs, RTs]).The approach used in the present paper is completely different: we start with a symmetry group of a manifold, more precisely, we are looking for a manifold whose symmetry group contains a given finite Weyl group.In [EM] Everitt and Maclachlan describe an algebraic framework for this approach, and in [KoSl] Kolpakov and Slavich explicitly construct an arithmetic hyperbolic 4-manifold with given finite isometry group.We use quite different technique based on mutations of quivers and diagrams defined by Fomin and Zelevinsky in the context of cluster algebras.This technique allows us to obtain manifolds having large symmetry groups and, at the same time, relatively small volume.
Mutation classes of quivers and diagrams of finite type are indexed by finite Weyl groups.Our interpretation of the result of [BM] is that, starting from a quiver or diagram G of finite type, one can present the corresponding finite Weyl group W as a quotient of some (usually infinite) Coxeter group W 0 (G) by the normal closure of additional relations W C (G).If this Coxeter group W 0 (G) is hyperbolic, then the Weyl group W acts properly on the quotient of the hyperbolic space by the group W C (G). Apriori, this quotient may have singular points.However, this is not the case.Theorem 6.2 (Manifold Property).The group W C (G) is torsion-free.
As a corollary, if the Coxeter group W 0 (G) is hyperbolic and cofinite, we obtain a hyperbolic manifold of finite volume, and the symmetry group of this manifold contains W .We list the manifolds that we found in this way in Tables 5.1 and 7.1.In dimension 4 we construct a manifold with Euler characteristic 2, i.e. it has the second minimal volume amongst all 4-dimensional manifolds.The symmetry group of this manifold contains the group Sym 5 ⋊ Z 2 as a subgroup.We note that, since this manifold has small volume and large symmetry group at the same time, it does not appear in [KoSl].
In [FeTu3] we generalized the results of [BM] to quivers and diagrams of affine type and certain other mutation-finite quivers and diagrams.In particular, every affine Weyl group also admits presentations of a similar type with slightly more complicated groups W C (G).Using this result, in Section 8.2 we provide the following theorem.
Theorem 8.1.The manifold property holds for quivers and diagrams of affine type.
For quivers and diagrams originating from unpunctured surfaces or orbifolds [FST,FeSTu3], as well as for exceptional mutation-finite quivers and diagrams [FeSTu1,FeSTu2], the construction in [FeTu3] also provides a group assigned to a mutation class.In contrast to the finite and affine cases, this group is not a Coxeter group, however, it also has presentations as a quotient of a Coxeter group W 0 (with more types of generators for W C required).
Using these presentations, for every quiver or diagram G of all types discussed above, we construct a CW-complex (actually being a quotient of the Davis complex of a Coxeter group W 0 (G)) with a proper action of the group W (G) = W 0 (G)/W C (G), where the group W (G) depends on the mutation class of G only.We also define mutations of these complexes, the mutations agree with quiver (or diagram) mutations.All the manifolds discussed above are partial cases of these complexes.
The paper is organized as follows.Section 2 contains basic definitions and some essential facts on Coxeter groups and quiver mutations.In Section 3 we recall the construction from [BM] assigning to a quiver of finite type a presentation of the corresponding Weyl group.For simplicity, in Sections 3-6 we restrict ourselves to simply-laced Weyl groups (and thus, we work with quivers rather than diagrams), we then consider general case in Section 7. In Section 4 we describe the construction highlighted above of various actions of a given finite Weyl group.In section 5 we first treat in full details our construction applied to the group A 3 ; then we list other geometric manifolds with Weyl group actions resulted from the algorithm described in Section 4. In particular, Table 5.1 contains the list of hyperbolic manifolds we obtain.In Section 6, after defining the Davis complex of a Coxeter group, we prove the Manifold Property.We also define mutations of quotients of Davis complexes agreeing with quiver mutations.Section 7 is devoted to generalizations of the results to all finite Weyl groups.Finally, in Section 8 we describe how the construction generalizes to affine Weyl groups and some other infinite groups related to quivers and diagrams of finite mutation type.

Quiver mutations and presentations of Coxeter groups
2.1.Coxeter groups.We briefly remind some basic properties of Coxeter groups.For the details see [D2].
where m ii = 1 and m ij ∈ N >1 ∪ ∞ for all i = j.Here m ij = ∞ means that there is no relation on s i s j .A pair (W, S) of a Coxeter group W and its set of generators S = {s 1 , . . ., s n } is called a Coxeter system.The cardinality n of S is a rank of the Coxeter system.An element of W is said to be a reflection if it is conjugated in W to an element of S.
A Coxeter system (W, S) may be depicted by its Coxeter diagram: • the vertices of the Coxeter diagram correspond to the generators s i ; • i-th vertex is connected to j-th by an edge labeled m ij with the following exceptions: -if m ij = 2 then there is no edge connecting i to j; -if m ij = 3 then the label 3 is usually omitted; -if m ij = ∞ the edge is drawn bold.Given a Coxeter system (W, S), one can write a symmetric n × n matrix M (W, S) = {M ij } as follows: An n × n matrix a ij is decomposable if there is a non-trivial partition of the index set as {1, . . ., n} = I ∪ J, so that a ij = a ji = 0 whenever i ∈ I, j ∈ J.A matrix is indecomposable if it is not decomposable.
2.1.2.Actions by reflections.For some Coxeter systems (W, S) the group W has a natural discrete action by reflections on a space of constant curvature (here by an action by reflections we mean an action where the reflections of W are represented by orthogonal reflections with respect to hyperplanes).
More precisely, there are three cases of interest: • M (W, S) is positive definite: then W is a finite group and W acts on a sphere S n−1 ; • M (W, S) is indecomposable and has signature (n−1, 0, 1): then W acts cocompactly on Euclidean space E n−1 ; • M (W, S) is indecomposable and has signature (n − 1, 1): then W acts on an (n − 1)-dimensional hyperbolic space H n−1 .See [V2] for the details.A fundamental domain of such an action is a connected component of the complement to the set of all hyperplanes fixed pointwise by reflections of the group.This fundamental domain is a polytope (which may be of infinite volume) with facets indexed by generators {s i } i=1,...,n and all dihedral angles equal π/m ij (if m ij = ∞ the corresponding facets do not intersect).Such polytopes are called Coxeter polytopes.
Remark 2.1.If a Coxeter group W is a finite or affine Weyl group, then the outer normal vectors to the facets of the fundamental polytope can be identified with the simple roots of the corresponding root system.
In the three cases listed above the group W is called geometric (note that our class of geometric Coxeter groups is wider than in [D2,Chapter 6]).In fact, every Coxeter group acts properly cocompactly by reflections on some specially constructed space [D1], we will discuss this action in Section 6.
2.2.Quiver mutations.Let Q be a quiver (i.e. a finite oriented multi-graph) containing no loops and no 2-cycles.If there are q arrows pointing from i-th vertex to the j-th, then we draw one arrow with a weight w ij = q.
2.2.1.Definition of a quiver mutation.For every vertex k of the quiver Q one can define an involutive operation µ k called mutation of Q in direction k.This operation produces a new quiver denoted by µ k (Q) which can be obtained from Q in the following way (see [FZ1]): • orientations of all arrows incident to the vertex k are reversed; • for every pair of vertices (i, j) such that Q contains arrows directed from i to k and from k to j the weight of the arrow joining i and j changes as described in Figure 2.1.Given a quiver Q, its mutation class is a set of all quivers obtained from Q by all sequences of iterated mutations.All quivers from one mutation class are called mutation-equivalent.
Quivers without loops and 2-cycles are in one-to-one correspondence with integer skew-symmetric matrices B = {b ij }, where b ij is the number of arrows from i-th vertex to j-th one.In terms of the matrix B the mutation µ k can be written as This transformation is called a matrix mutation.
Remark 2.2.The procedure of matrix mutation is well-defined for more general case of skew-symmetrizable integer matrices.Matrix mutations for non-skew-symmetric matrices correspond to diagram mutations (see Section 7 for the definitions and generalizations of the results to the case of diagram mutations).[FeSTu1] that each quiver of finite mutation type is either of order 2, or a quiver arising from a triangulated surface (see [FST] for details), or mutation-equivalent to one of the 11 exceptional quivers.
In this paper we deal with quivers of finite mutation type only.

Presentations of Coxeter groups arising from quiver mutations
In [BM] Barot and Marsh show that for each quiver of finite type there is a way to explicitly construct the corresponding Weyl group.
3.1.Construction of the group by a quiver.Let Q be an orientation of a simply-laced Dynkin diagram with n nodes, let W be the corresponding finite Coxeter group, and Q 1 be any quiver mutationequivalent to Q. Denote by W (Q 1 ) the group generated by n generators s i with the following relations: (R1) s 2 i = e for all i = 1, . . ., n; (R2) (s i s j ) mij = e for all i, j, where m ij = 2 if i and j are not connected; 3 if i and j are connected by an arrow.
(R3) (cycle relations) for every chordless oriented cycle C given by we take the relation Note that for a cycle of length d we have a choice of d relations of type (R3), we take any one of them.
It is shown in [BM, Theorem A] that the group W (Q 1 ) does not depend on the choice of a quiver in the mutation class of Q.In particular, it is isomorphic to the initial Coxeter group W .
Remark 3.1.The results of [BM] are proved in more general settings, we come to the general case in Section 7.
3.2.Generators of W (Q 1 ).Generators of W (Q 1 ) satisfying relations (R1)-(R3) can be expressed in terms of standard generators of W .It is shown in [BM] that these generators of W (Q 1 ) can be found inductively in the following way: 1. for Q 1 = Q the generators {s i } are standard Coxeter generators (indeed, Dynkin diagrams contain no cycles, so, we will get no cycle relations and obtain standard presentation of the Coxeter group).2. Let {s i } be the generators of W (Q 1 ), and let satisfying relations (R1)-(R3) are

Construction of various actions of W
As before, let Q be an orientation of a Dynkin diagram, let W be the corresponding Weyl group.Choose any quiver Q 1 mutation-equivalent to Q, and denote by W 0 (Q 1 ) the group defined by relations of type (R1) and (R2) only (note that W 0 (Q 1 ) is a Coxeter group).
Let C 1 , . . ., C p be a collection of all cycle relations for Q 1 (we take one relation for each chordless oriented cycle).Denote by W C (Q 1 ) the normal closure of the union of the elements C The group W 0 (Q 1 ) depends on the quiver Q 1 chosen in the mutation class of Q.Moreover, depending on the choice of Q 1 , the group W 0 (Q 1 ) may be finite or infinite.As a consequence, it may act naturally by reflections on different spaces (sphere, Euclidean space, hyperbolic space or, in general, the Davis complex of the group In Section 6 we show that the group W C (Q 1 ) is torsion-free (we call this Manifold Property).In particular, this implies that if the Coxeter group W 0 (Q 1 ) acts properly on a space X = E n or H n by isometries with a fundamental domain of finite volume, then the quotient X/W C (Q 1 ) is a finite volume manifold with a symmetry group containing W .In the next section we show examples of construction of hyperbolic manifolds in this way, as well as of actions of Weyl groups of type D on flat tori.

Examples of manifolds constructed via quiver mutations
5.1.Action of A 3 on the flat 2-torus.We now present the first (and the easiest) example showing how different actions of the same Coxeter group may by produced via mutations of quivers.5.1.1.Action on the sphere.We start with the quiver Q of type A 3 (see Fig. 5.1(a),(b)).The corresponding Dynkin diagram of type A 3 determines a finite Coxeter group W acting on the 2-dimensional sphere by reflections: The fundamental domain of this action is a spherical triangle with angles ( π 3 , π 3 , π 2 ), see Fig. 5.1(c) for the stereographic projection image of the action and the fundamental domain.on the sphere (in stereographic projection): a sphere is tiled by 24 copies of (spherical) triangles with angles ( π 2 , π 3 , π 3 ).
5.1.2.Action on the flat 2-torus.Applying the mutation µ 2 to the quiver Q, we obtain a quiver Q 1 (see Fig. 5.2(a)).The group = e is the group generated by reflections in the sides of a regular triangle on Euclidean plane (the corresponding Dynkin diagram is of type A 2 ), see Fig. 5.2(b).Note that the plane is tessellated by infinitely many copies of the fundamental domain, each copy F g labeled by an element g ∈ W 0 (Q 1 ).To see the action of the initial group A 3 = W = W (Q 1 ) we need to take a quotient by the cycle relation which means that in the tessellated plane we identify F e with the fundamental triangle labeled by the (t 1 t 2 t 3 t 2 ) 2 .Now, let us see what does the transformation T 1 = (t 1 t 2 t 3 t 2 ) 2 mean geometrically.The element t 1 acts on the plane as the reflection in the side l 1 of F e , t 2 t 3 t 2 is the reflection with respect to the line l 232 parallel to l 1 and passing through the vertex of F e not contained in l 1 .Denote by d the distance between l 1 and l 232 .Then the transformation (t 1 t 2 t 3 t 2 ) is a translation of the plane by the distance 2d in the direction orthogonal to l 1 .Hence, the element T 1 = (t 1 t 2 t 3 t 2 ) 2 is a translation by 4d in the direction orthogonal to l 1 .
Clearly, e = t 2 (t This implies that we also need to take a quotient of the plane by the actions of the elements T 3 = (t 2 t 1 t 2 t 3 ) 2 and T 2 = (t 3 t 1 t 3 t 2 ) 2 , i.e. by the translations by 4d in directions orthogonal to the lines l 2 and l 3 .The quotient of the plane by the group W C (Q 1 ) generated by three translations T 1 , T 2 and T 3 is a flat torus, see Fig. 5.2(c).Thus, we obtained an action of the quotient group A 3 = W = W (Q 1 ) on the torus: the fundamental domain of this action is a regular triangle, 24 copies of this triangle tile the torus (as 24 copies tile the hexagon).
Remark 5.1.The action described above is not an action by (topological) reflections: the fixed sets of the generators t i are connected, so they do not divide the torus into two connected components (cf.[D2, Chapter 10]).5.1.3.From the sphere to the torus: ramified covering.Now, we will show how the action of A 3 on the torus is related to the initial action on the sphere.
Recall from Section 3.2 that the presentation for W (Q 1 ) is obtained from the presentation for W (Q) by the following change of generators (s 1 , s 2 , s 3 ) → (t 1 , t 2 , t 3 ) where t 1 = s 2 s 1 s 2 , t 2 = s 2 , t 3 = s 3 .
In the above sections we considered the action of the s-generators on the sphere and of the t-generators on the plane and the torus.However, the t-generators may be also considered directly on the sphere.Namely, let s 1 , s 2 , s 3 be the reflections with respect to the corresponding sides l 1 , l 2 , l 3 of the fundamental spherical triangle ∆ s , see Fig. 5.3(a).Then the elements t 2 and t 3 are reflections with respect to l 2 and l 3 respectively.The element t 1 is also a reflection as it is conjugate to the reflection s 1 ; more precisely, t 1 is a reflection with respect to the (spherical) line l 212 (see Fig. 5.3(b)).
Consider the triangle ∆ t bounded by l 212 , l 2 , and l 3 (there are 8 possibilities to choose such a triangle, see Remark 5.2 for the justification of our choice).
In contrast to the triangle ∆ s bounded by l 1 , l 2 , l 3 , the triangle ∆ t is not a fundamental triangle for the action: ∆ t has angles ( π 3 , π 3 , 2π 3 ), and the group generated by t 2 and t 3 has order 6, but the whole neighborhood of the vertex p ∈ l 2 ∩ l 3 is tiled by 3 copies of ∆ t only.
To resolve this inconsistency we consider a 2-sheet covering of the sphere branched in p and in all W -images of p (the four bold points in Fig. 5.3(b)).Clearly, the covering space is a torus (moreover, one can easily see that the covering is given by Weierstrass ℘-function).It is also not difficult to see that the action of A 3 on this covering torus is exactly the same as its action on the torus in Section 5.1.2:the covering torus consists of 24 triangles, and every vertex is incident to exactly six triangles.Remark 5.2 (On the choice of the triangle ∆ t ).Consider the outer normals v 1 , v 2 , v 3 to the sides of ∆ s (cf.Remark 2.1).To get the outer normals u 1 , u 2 , u 3 to the sides of ∆ t we use the following rule: Here s k (v i ) is the vector obtained from v i by the reflection s k .
5.1.4.Back from the torus to the sphere.Now we will obtain the action of A 3 on the sphere from its action on the torus.Let ∆ t be the fundamental triangle for the action of A 3 on the flat torus (see Fig. 5.4(a)).Let π be the canonical projection from Euclidean plane to the torus (corresponding to the action of W C (Q 1 ) on the plane), denote by ∆ t any preimage of ∆ t under π.The generators t 1 , t 2 , t 3 can be thought as reflections with respect to the sides of ∆ t .The generators s i can be obtained from the generators t i as follows: These are reflections with respect to the sides of the dashed domain ∆ s (see Fig. 5.4(b)), where ∆ s is an unbounded strip on the Euclidean plane.
We now glue a manifold from 24 copies of the strip (according to the group action, as we glued 24 triangles above), denote by X the space obtained via this gluing.Note that due to the cycle relations four strips attached one to another along the "long" (unbounded) sides will result in an (unbounded) cylinder.The manifold X consists of exactly 6 = 24 : 4 cylinders.Compactifying X by adding a limit point at the end of each of the 6 cylinders, we get exactly the tiling shown in Fig. 5.1(c).Note that the group W 0 (Q 1 ) (i.e.Coxeter group without cycle relations) is the affine Coxeter group A n−1 , thus, a group acting on the Euclidean space E n−1 .A fundamental domain of this action is a Coxeter simplex ∆ with the Coxeter diagram A n−1 .The generators for this action are the reflections t 1 , . . ., t n with respect to the facets Π 1 . . ., Π n of this fundamental simplex.
Our next aim is to take a quotient by the cycle relation The element (t 2 t 3 . . .t n−1 t n t n−1 . . .t 3 t 2 ) is a conjugate of t n , so it is a reflection with respect to the hyperplane Π parallel to Π 1 and passing though the vertex of ∆ opposite to Π 1 .If d is the distance between the hyperplanes Π and Π 1 , then the element (t 1 t 2 t 3 . . .t n−1 t n t n−1 . . .t 3 t 2 ) is a translation by 2d in the direction orthogonal to Π 1 , and its square 2 is a translation by 4d.So, we need to take the quotient by the action of this translation.The element They all are translations by 4d in the direction orthogonal to Π k .The group generated by T 1 , . . ., T n is a free abelian group of rank n − 1.Hence, taking a quotient of E n−1 by the translations T 1 , . . ., T n we get an (n − 1)-dimensional torus T n−1 .The action of W 0 (Q 1 ) on E n−1 turns into a faithful action of D n = W (Q 1 ) on the torus T n−1 .
Remark 5.3.The Coxeter diagram A n is the unique affine Dynkin diagram containing a cycle.This implies that our construction gives no further examples of actions on non-trivial quotients of Euclidean space.
5.3.Actions on hyperbolic manifolds.We now consider the case when the group W 0 (Q 1 ) acts on a hyperbolic space, and hence, the group W = W (Q 1 ) acts on its quotient X = H n /W C (Q 1 ) (recall that W C (Q 1 ) is the normal closure of all cycle relations).As it is shown below (Theorem 6.2, Corollary 6.3), in this case X is always a manifold.In this section we list all known actions on hyperbolic manifolds obtained in this way (Table 5.1).
Notation.In Table 5.1 we omit orientations of arrows of quivers Q and Q 1 .Here arrows of Q may be oriented in arbitrary way, and the only requirement on the orientations of arrows of Q 1 is that all chordless cycles are oriented.
The underlying graphs of Q and Q 1 are the Coxeter diagrams of fundamental polytopes of the action of W and W 0 (Q 1 ) on the sphere and the hyperbolic space respectively.All the volumes except the last two are computed in [JKRT], the Euler characteristic (and thus the volume) of the fundamental polytope of the action of D 8 was computed by the program CoxIter written by Rafael Guglielmetti.The volume of the fundamental polytope of the action of A 7 is not known to the authors.Euler characteristic is computed for all even-dimensional manifolds.The precise value of volume of X for four-dimensional X is equal to 4π 2 χ(X)/3, and for six-dimensional X equals −8π 3 χ(X)/15.
Remark 5.4.The manifold X obtained in the case W = A 4 is the manifold M 5 studied in [MPR], it has minimal known volume amongst 5-cusped hyperbolic 3-manifolds.It can be obtained via a ramified 2-fold covering over the 3-sphere S 3 branching along 1-skeleton of a 4-simplex.This manifold can be also obtained as the complement of the five-chain link in the 3-sphere.
Remark 5.5.The symmetry group of X always contains W . Further, if the underlying graph of the quiver Q 1 has symmetries, then the fundamental polytope of the action of W (Q 1 ) on X also has these symmetries, and thus the symmetry group of X also contains a semidirect product of W and the group of symmetries of the graph.In particular, one can note from Table 5.1 that the underlying graph of Q 1 always has an additional symmetry of order 2, therefore we actually get the group W ⋊ Z 2 acting on X.Moreover, in the cases W = D 4 , A 7 or D 8 the symmetry group of the fundamental polytope is Z 2 × Z 2 , which results in the action of the group W ⋊ (Z 2 × Z 2 ) on the manifold X.
1.The list in Table 5.1 is complete according to the following.Any two nodes of any quiver of finite type are connected by at most one arrow, therefore all dihedral angles of the corresponding fundamental polytope of the group action are either π/2 or π/3.All hyperbolic Coxeter polytopes satisfying such property are classified by Prokhorov in [P].One can see that the only Coxeter diagrams whose orientations are quivers of finite type are those listed in Table 5.1.2. In all cases the fundamental domain of the action W (Q 1 ) on X is a non-compact hyperbolic polytope, i.e. the manifold X has cusps.3. The combinatorial type of the fundamental domain in all but last two cases is a simplex.In the last two cases it is a pyramid over a product of two simplices.

Actions on quotients of Davis complex
In this section we first recall the construction of Davis complex Σ (W, S) for each Coxeter system (W, S).
Then, given a Dynkin quiver Q and a quiver Q 1 from the same mutation class, we consider the Davis complex Σ(W 0 (Q 1 )) for the group W 0 (Q 1 ) together with the action of the Coxeter group W = W (Q) on the corresponding quotient space X = Σ(W 0 (Q 1 ))/W C (Q 1 ) and prove the Manifold Property for X (which states that W C (Q 1 ) is torsion-free).In particular, we prove that all quotients listed in Table 5.1 are hyperbolic manifolds.6.1.Davis complex.For any Coxeter system (W, S) there exists a contractible piecewise Euclidean cell complex Σ (W, S) (called Davis complex) on which W acts discretely, properly and cocompactly.The construction (based on results of Vinberg [V1]) was introduced by Davis [D1].In [M] Moussong proved that this complex yields a natural complete piecewise Euclidean metric which is CAT (0).We first give a brief description of this complex following [NV], and then follow [FeTu2] to define convex polytopes in Σ (W, S).
6.1.1.Construction of Σ (W, S).For a finite group W the complex Σ (W, S) is just one cell, which is obtained as a convex hull K of W -orbit of a suitable point p in the standard linear representation of W as a group generated by reflections.The point p is chosen in such a way that its stabilizer in W is trivial and all the edges of K are of length 1.The faces of K are naturally identified with Davis complexes of the subgroups of W conjugate to standard parabolic subgroups.
If W is infinite, the complex Σ (W, S) is built up of the Davis complexes of maximal finite subgroups of W glued together along their faces corresponding to common finite subgroups.The 1-skeleton of Σ (W, S) considered as a combinatorial graph is isomorphic to the Cayley graph of W with respect to the generating set S.
The action of W on Σ (W, S) is generated by reflections.The walls in Σ (W, S) are the fix point sets of reflections in W .The intersection of a wall α with cells of Σ (W, S) supplies α with a structure of a piecewise Euclidean cell complex with finitely many isometry types of cells.Walls are totally geodesic: any geodesic joining two points of α lies entirely in α.Since Σ is contractible and CAT (0), any two points of Σ can be joined by a unique geodesic.
Any wall divides Σ (W, S) into two connected components.All the walls decompose Σ (W, S) into connected components which are compact sets called chambers.Any chamber is a fundamental domain of the W -action on Σ (W, S).The set of all chambers with appropriate adjacency relation is isomorphic to the Cayley graph of W with respect to S.
In what follows, if W and S are fixed, we write Σ(W ) or Σ instead of Σ (W, S).
Remark 6.1.If the group W acts cocompactly by reflections on sphere, Euclidean or hyperbolic space X, then the Davis complex Σ is quasi-isometric to X.
6.1.2.Convex polytopes in Σ.For any wall α of Σ we denote by α + and α − the closures of the connected components of Σ \ α, we call these components halfspaces.
A convex polytope P ⊂ Σ is an intersection of finitely many halfspaces P = n i=1 α + i , such that P is not contained in any wall.Clearly, any convex polytope P ⊂ Σ is a union of closed chambers.
In what follows by writing P = n i=1 α + i we assume that the collection of walls α i is minimal: for any j = 1, . . ., n we have A facet of P is an intersection P ∩ α i for some i ≤ n.For any I ⊂ {1, . . ., n} a set f = i∈I α i ∩ P is called a face of P if it is not empty.
We can easily define a dihedral angle formed by two walls: if α i ∩ α j = ∅ there exists a maximal cell C of Σ intersecting α i ∩ α j .We define the angle ∠ (α i , α j ) to be equal to the corresponding Euclidean angle formed by α i ∩ C and α j ∩ C. Clearly, any dihedral angle formed by two intersecting walls in Σ is equal to kπ/m for some positive integers k and m.A convex polytope P is called acute-angled if each of the dihedral angles of P does not exceed π/2.
A convex polytope P is called a Coxeter polytope if all its dihedral angles are integer submultiples of π.In particular, a fundamental domain of any reflection subgroup of W is a Coxeter polytope in Σ.Conversely, any Coxeter polytope in Σ is a fundamental chamber for the subgroup of W generated by reflections in its walls (see e.g.[D2, Theorem 4.9.2]).6.2.Manifold Property.Let us briefly recall the procedure described in Section 4: 1. Start with a Dynkin quiver Q with corresponding Weyl group W .Let Q 1 be any quiver mutationequivalent to Q. 2. Consider the Coxeter group W 0 (Q 1 ) defined by the relations (R1) and (R2) (its Coxeter diagram coincides with the underlying graph of the quiver Q 1 ).3. Let Σ(W 0 (Q 1 )) be the Davis complex of the group W 0 (Q 1 ).By construction, W 0 (Q 1 ) acts on Σ(W 0 (Q 1 )) by reflections.4. Let C 1 , . . .C p be the collection of all defining cycle relations for Q 1 , define W C (Q 1 ) being the normal closure of all these relations in W 0 (Q 1 ).

Define
Then W = W (Q 1 ) acts properly on X(Q 1 ).So, for each quiver in the mutation class we assigned an action of W on the space X(Q 1 ) (the latter heavily depends on the quiver Q 1 ).Theorem 6.2 (Manifold property).The group Let F be a fundamental domain for the action of W (Q 1 ) on Σ(W 0 (Q 1 )).A star Star (f ) of a face f of F is the union of all fundamental domains F i of Σ containing f : We will prove the following statement which is equivalent to the Manifold Property (cf.[EM,Corollary 2]).Theorem 6.4.Let F be a fundamental domain for the action of W (Q 1 ) on Σ(W 0 (Q 1 )).Then for any face f of F the interior ( Star (f )) 0 of the star of f in X(Q 1 ) is isometric to the interior of the star of Equivalently, if a face f of F is contained in q fundamental domains of the action of W 0 (Q 1 ) on Σ(W 0 (Q 1 )), then f is also contained in q fundamental domains of the action of W (Q 1 ) on X(Q 1 ).
To prove the Manifold Property we will use the following observation.
Lemma 6.5.Let ti and tj be two generating reflections of W 0 (Q 1 ), and let t i and t j be their images in Then the orders of the elements ti tj and t i t j coincide.
Proof.The dihedral group generated by t i and t j is a quotient of the dihedral group generated by ti and tj .The order of the latter group equals 6 or 4, therefore, if we assume that the two groups are distinct, the order of the former equals two, i.e. t i = t j , which is impossible since a Coxeter group W of rank n cannot be generated by less than n reflections (see e.g.[FeTu2, Lemma 2.1]).
Proof of Theorem 6.4.Let f be a face of F .Denote by I ⊂ {1, . . ., n} the index set such that f is the intersection of the walls fixed by { ti } i∈I .To prove Theorem 6.4 (and thus the Manifold Property) it is sufficient to prove that the group T I ⊂ W 0 (Q 1 ) generated by { ti } i∈I has the same order as the group The group T I can be considered as a group generated by reflections in the facets of a spherical Coxeter (k − 1)-simplex P with dihedral angles π/m ij , where m ij is the order of ti tj .Furthermore, since every t i ∈ W = W (Q) is a reflection on the X(Q) = S n−1 , the group T I can also be considered as a group generated by reflections in the facets of a spherical (k − 1)-simplex P .The simplex P may not be Coxeter, however, by Lemma 6.5, its dihedral angles are equal to πp ij /m ij for some p ij being coprime with m ij .Since m ij = 2 or 3, this implies that either p ij = 1, or p ij /m ij = 2/3.According to [Fe], this means that reflections in the facets of P and P generate isomorphic groups.6.3.Mutation of complexes.In this section we describe how to obtain X(Q 2 ) from X(Q 1 ), where Q 2 = µ k (Q 1 ).The construction generalizes the examples shown in Sections 5.1.2and 5.1.4.
Step 1.Let s 1 , . . ., s n be the generators of W corresponding to Q 1 (i.e., if {s i } with relations (R1) and (R2) are generating reflections of W 0 (Q 1 ) composing a Coxeter system, and the {s i } with relations (R1)-(R3) are their images under canonical homomorphism W 0 (Q 1 ) → W 0 (Q 1 )/W C (Q 1 )), let t 1 , . . ., t n be the generators corresponding to Q 2 .As it is shown in [BM], t i can be obtained from s i as follows Denote by π j , j = 1, 2, the canonical projection Σ(W 0 (Q j )) → X(Q(j)).According to the construction, a fundamental chamber F j ⊂ X(Q j ) of the action of W on X(Q j ) is an image of a fundamental chamber F j ⊂ Σ(W 0 (Q j )) under the projection.The elements s i act on Σ(W 0 (Q 1 )) by reflections in the facets of F 1 , let walls α1 , . . ., αn ⊂ Σ(W 0 (Q 1 )) be the mirrors of the reflections s 1 , . . ., s n .Similarly, the elements t i are reflections in the facets of F 2 , let walls β1 , . . ., βn ⊂ Σ(W 0 (Q 2 )) be the mirrors of the reflections t 1 , . . ., t n .
Step 2. Consider the action of the element t i on Σ(W 0 (Q 1 )).It is a reflection with respect to a wall γi satisfying γi = s k (α i ) if there is an arrow from i to k in Q 1 ; αi otherwise.
Step 3. The fundamental chamber F 1 is an intersection of n halfspaces bounded by walls { αi }.Denote these halfspaces by α+ i , i.e.F 1 = α+ i .Note that the dihedral angles of a polytope F 1 equal π/m ij , where m ij is the order of s i s j in W 0 (Q 1 ).In particular, as all m ij = 2 or 3, every two facets of F 1 have a non-empty intersection.
Step 4. Consider the domain H ⊂ Σ(W 0 (Q 1 )) defined as follows: i , where the walls γi are defined in Step 2 and the half-spaces γ+ i , i = k, are chosen by Note that H is well-defined, i.e.H indeed has exactly n facets.This follows from the fact that the facet αk ∩ F 1 of F 1 (which is also a facet of H) has exactly n − 1 facets αk ∩ αi ∩ F 1 , and by construction of H all these faces of F 1 are also faces of H.
Step 5.The domain H is a convex (possibly, non-compact) polytope in Σ(W 0 (Q 1 )) with n facets.The reflections with respect to the facets of H are precisely t i , they generate the group W 0 (Q 1 ), however, in general, H is not a fundamental domain of the action of W 0 (Q 1 ) on Σ(W 0 (Q 1 )).Step 6.Now, we will use H to construct a new cell complex Y (such that the group W acts on Y with fundamental chamber H).
Consider |W | copies of the polytope H, labeled by the elements of W (denote by H w the polytope corresponding to w ∈ W ).
Glue the polytope H w to the polytope H wti along the facets γi of H w and H wti so that the reflection in the common facet swaps the polytopes (and takes the facet γj of H w to the facet of H wti having the same number).Denote by Y the space obtained by the gluing.Note that by "reflection" here we mean a map from one polytope to another, this may not extend to a reflection on the whole Y Remark 5.1).
Step 7. Denote by m ij the order of the subgroup t i , t j ⊂ W (note that the Coxeter group -First, we complete the fundamental domain H (obtaining a completed fundamental chamber H ′ ) in the following way.For every pair {i, j} ⊂ {1, . . .n} we add an (n − 2)-dimensional face γi ∩ γj if it is not a face of H yet; we define the dihedral angle composed by facets γi ∩ H and γj ∩ H of H ′ as π/m ij .Then, for each finite subgroup t i | i ∈ I ⊂ W 0 (Q 2 ) (where I ⊂ {1, . . ., n} is an index set) we add an (n − |I|)-dimensional face i∈I γi if it is not a face of H yet.The procedure is well-defined since any subgroup of a finite group is finite.
The resulting polytope H ′ can be identified with a fundamental chamber of Σ(W 0 (Q 2 )), i.e. with F 2 .-Now, we do this for each H w , w ∈ W (a new face i∈I γi of H w is identified with the similar face of H wtj if j ∈ I).The procedure results in a new complex Y ′ .Step 8.As H ′ can be identified with F 2 , and Y ′ is obtained from copies of H ′ by gluings along the action of W (see Step 6), the complex Y ′ can be identified with

General case
In all the considerations above we started with a Dynkin quiver Q.In this section we consider mutations of diagrams instead of quivers, which allows us to deal with non-simply-laced Dynkin diagrams.7.1.Mutations of diagrams.Recall that quivers are in one-to-one correspondence with skew-symmetric integer matrices.An integer n × n matrix B is skew-symmetrizable if there exists a diagonal n × n matrix D = {d 1 , . . ., d n } with positive integer entries such that the product BD is skew-symmetric, i.e., b ij d j = −b ji d i .
Given a skew-symmetrizable matrix B one obtains a diagram G = G(B) in the following way: G is an oriented labeled graph with vertex set 1, . . ., n, where vertex i is connected to vertex j by an arrow labeled by One diagram may correspond to several distinct matrices.Any diagram G constructed by a skewsymmetrizable matrix satisfies the following property: for any chordless cycle in G the product of labels along this cycle is a perfect square (cf.[K,Exercise 2.1]).
Since skew-symmetric matrices are also skew-symmetrizable, we may understand quivers as a partial case of diagrams: to make a diagram out of a quiver one needs to square all the labels.
A mutation µ k of a diagram G is defined similarly to the mutation of quivers [FZ2]: • orientations of all arrows incident to a vertex k are reversed; • for every pair of vertices (i, j) such that G contains arrows directed from i to k and from k to j the weight of the arrow joining i and j changes as described in Figure 7.1.As for quivers, a diagram is of finite type if it is mutation-equivalent to an orientation of a Dynkin diagram.7.2.Generalization of the construction to non-simply-laced diagrams of finite type.The construction is generalized straightforwardly, but the definition of the group defined by a diagram is a bit longer than one in the simply-laced case.
Let G be an orientation of a Dynkin diagram with n nodes, let W be the corresponding finite Weyl group, and let G 1 be any diagram mutation-equivalent to G. Denote by W (G 1 ) the group generated by n generators s i with the following relations: (R1) s 2 i = e for all i = 1, . . ., n; (R2) (s i s j ) mij = e for all i, j, not joined by an edge labeled by 4, where 2 if i and j are not connected; 3 if i and j are connected by an edge.4 if i and j are connected by an edge labeled by 2; 6 if i and j are connected by an edge labeled by 3.
(R3) (cycle relation) for every chordless oriented cycle C given by i 0 , where the indices are considered modulo d; now for every l such that t(l) < 4, we take the relation (this form of cycle relations was introduced by Seven in [Se]).
According to [BM], all the cycle relations for a given chordless cycle follow from one with m(l) = 2 (such a relation always exists).Thus, as for quivers, we may take exactly one defining relation per cycle.
It is shown in [BM] that the group W (G 1 ) does not depend on the choice of a diagram in the mutation class of G.In particular, it is isomorphic to the initial Coxeter group W .
The further part of the construction generalizes to the diagram settings verbatim.We obtain three manifolds, see Table 7.1 Table 7.1.Actions on hyperbolic manifolds, non-simply-laced case.
The manifold X constructed for the group B 4 coincides with the manifold constructed in Section 5.3 for the group D 4 (see Table 5.1).
Remark 7.2.In contrast to the simply-laced case, we do not know whether the list of hyperbolic manifolds (Table 7.1) that can be obtain by our construction is complete: there is no known classification of finite volume hyperbolic Coxeter polytopes with angles π/2, π/3, π/4, so potentially the construction may provide other examples of finite volume hyperbolic manifolds.
Remark 7.3.The proof of the Manifold Property works for diagrams, but we need to update the proof of Lemma 6.5.
Let us proceed by induction on the number of mutations required to obtain G 1 from G. Assume that the lemma holds for a diagram G 2 mutation-equivalent to G, and we want to deduce the lemma for G 1 = µ k (G 2 ).In the notation of Lemma 6.5 (after substituting Q by G) this means the following: if we denote by ũi , ũj , ũk ∈ W 0 (G 2 ) the reflections corresponding to nodes i, j, k of G 2 and by u i , u j , u k ∈ W (G 2 ) = W their projections to W , we want to deduce the equality of orders of elements ti tj ∈ W 0 (G 1 ) and t i t j ∈ W (G 1 ) = W from the equality of the orders of ũi ũj ∈ W 0 (G 2 ) and u i u j ∈ W (G 2 ) = W .
If one of i, j (say, i) equals k the statement is obvious: in this case the label of the arrow between i and j is the same in G 1 and G 2 (and thus the orders of ti tj ∈ W 0 (G 1 ) and ũi ũj ∈ W 0 (G 2 ) are the same), and t i t j = (u i u j ) ±1 (since either t j = u j or t j = u i u j u i ), so the orders of t i t j and u i u j are also the same.
If the other case we can restrict our consideration to the subdiagrams of G 1 and G 2 spanned by three vertices i, j, k.A subdiagram of a diagram of finite type also has finite type [FZ2], so if it is not simplylaced then G 1 (and G 2 ) is mutation-equivalent to one of the three diagrams B 3 , B 2 + A 1 and G 2 + A 1 .Now, a short straightforward check verifies the lemma.W 0 (G 1 ) = t 1 , t 2 , t 3 | t 2 i = (t 1 t 2 ) 3 = (t 2 t 3 ) 4 = (t 1 t3) 4 = e acts on the hyperbolic plane by reflections, and the fundamental domain of this action is a triangle F with angles ( π 3 , π 4 , π 4 ).We need to take a quotient of this action by the normal closure of the element (t 1 t 2 t 3 t 2 ) 2 ∈ W 0 (G 1 ).As we expect from Manifold Property, we will get some finite volume hyperbolic surface S tiled by 48 images of the triangle F .It is easy to see that S is a genus 3 surface.
Similarly to Section 5.1.3describing A 3 case, one can see that S can be obtained as a ramified degree 4 covering of the sphere branching at six points composing the orbit of the "central point" (see Fig. 7.3(c)).

Infinite groups
8.1.Group constructed by a diagram.The result of [FeTu3] allows to generalize the construction to infinite groups.More precisely, it is shown in [FeTu3] that to a diagram G of finite mutation type that either is exceptional (see [FeSTu1,FeSTu2]) or arises from a triangulation of a unpunctured surface or orbifold (see [FST,FeSTu3]) one can assign a Coxeter group W 0 (G) (using exactly the same rule as above).
Then one takes a quotient group imposing relations of the following types: (R3) cycle relations; (R4) additional affine relations; (R5) additional handle relations; (R6) additional X 5 -relations.All these relations are of the type (r i r j ) m = e, where r i and r j are some conjugates of the generating reflections, see [FeTu3] for details.
As before, we define W C (G) as the normal closure of all relations of types (R3)-(R6).It is shown in [FeTu3] that the group W (G) = W 0 (G)/W C (G) does not depend on the choice of G in a given mutation class.
This implies that we can repeat verbatim the construction of actions on factors of the Davis complex Σ(W 0 (G)), so that we get an action of an (infinite) group W = W 0 (G)/W C (G) on the quotient space X(G) = Σ(W 0 (G))/W C (G).Furthermore, as before, a mutation of G induces a mutation of X(G).Mutation classes of diagrams of affine type may be rather complicated (see [H] for the detailed description of mutation classes of non-exceptional diagrams of finite and affine type).In particular, a diagram G in the mutation class may contain one arrow labeled by 4 (this corresponds to two reflections in a Coxeter system of W 0 (G) generating an infinite dihedral group, or, equivalently, to two non-intersecting facets of a fundamental chamber of Σ(W 0 (G)).
Nevertheless, applying considerations from Section 6.3 and Remark 7.3, we obtain the Manifold Property for diagrams of affine type.Remark 8.2.We are unable to drop the assumption that G is of affine type.Our proof of Manifold Property uses an induction on the number of mutations one needs to perform to obtain a diagram without oriented cycles (such diagrams are called acyclic, and diagrams from their mutation classes are mutationacyclic).If G is not of finite or affine type, then either G is not mutation-acyclic, or it is not mutation-finite (see [FST,FeSTu1,FeSTu2]).

Figure 2 . 1 .
Figure 2.1.Mutations of quivers.The sign before c (resp., d) is positive if the three vertices form an oriented cycle, and negative otherwise.Either c or d may vanish.If ab is equal to zero then neither value of c nor orientation of the corresponding arrow does change.

FeF
Figure 5.2.(a) mutation µ 2 ; (b) action of W 0 (Q 1 ) = A 2 on the plane; (c) action of W = W (Q 1 ) = A 3 on the torus: 24 regular triangles tile the hexagon whose opposite sides are identified.

Figure 5
Figure 5.3.Construction of the ramified covering

5. 2 .
Actions of D n on the flat (n − 1)-torus.The next series of examples is provided by the finite Coxeter group W = D n , n > 3 (see Fig.5.5(a) for the Dynkin diagram), which naturally acts on (n − 1)dimensional sphere (for n = 3 the Dynkin diagram and Weyl group D 3 coincides with A 3 , so the example from the previous section can be considered as a partial case).It is easy to check that the quiver Q of type D n (see Fig.5.5(b)) is mutation-equivalent to the oriented cycle Q 1 shown in Fig.5.5(c) (in contrast to A 3 -example, one needs more than one mutation for this).

Figure 7 . 1 .
Figure 7.1.Mutations of diagrams.The sign before √ c (resp., √ d) is positive if the three vertices form an oriented cycle, and negative otherwise.Either c or d may vanish.If ab is equal to zero then neither value of c nor orientation of the corresponding arrow does change.

Figure 7
Figure 7.2.(a) A diagram of type B 3 and its mutation µ 2 ; (b) action of B 3 on the sphere, in s-generators (stereographic projection); (c) action of B 3 on the sphere in tgenerators: bold points are the branching points for the covering (one more branching point at infinity)

8. 2 .
Diagrams of affine type.A diagram G is called a diagram of affine type if G is mutation-equivalent to an orientation of an affine Dynkin diagram different from an oriented cycle.In particular, if G is of affine type, the group W = W (G) is an affine Coxeter group.
Theorem 8.1.If G is a diagram of affine type then the group W C (G) is torsion-free.