Drinfeld-Manin solutions of the Yang-Baxter equation coming from cube complexes
DDrinfeld-Manin solutions of the Yang-Baxterequation coming from cube complexes
Alina VdovinaJuly 3, 2020
Abstract
The most common geometric interpretation of the Yang-Baxter equa-tion is by braids, knots and relevant Reidemeister moves. So far, cubeswere used for connections with the third Reidemeister move only. Wewill show that there are higher-dimensional cube complexes solving the D -state Yang-Baxter equation for arbitrarily large D . More precisely, weintroduce explicit constructions of cube complexes covered by products of n trees and show that these cube complexes lead to new solutions of theYang-Baxter equations. The Yang-Baxter equations appear in many fields of mathematics and theoret-ical physics: they are defined in many ways, possess many interpretations andoccur in many contexts.Often in the literature, the Yang-Baxter equation is called “triangular” —see, for example [6] — but we will show that a very natural framework liesin those n -dimensional cube complexes whose universal cover is a Cartesianproduct of n trees. Our main result is that each n -dimensional cube complexwhose universal cover is a Cartesian product of n trees induces a Drinfeld-Maninsolution of the Yang-Baxter equation. The following definition distills the resultof analysing over 30 years’ worth of papers. Definition 1.
Let X be a (non-empty) set, and R : X → X be a bijectiongiven by R ( x, y ) = ( u, v ) . We call R a Drinfeld-Manin solution of the Yang-Baxter equation if R R R = R R R on X , where R ij means R acting on i -th and j -th component of X . This definition is quite close to what is called a set-theoretical solution ora
Yang-Baxter map in the literature; see, [5, 19, 4]. See also [3] for the mostrecent account of the subject. 1 a r X i v : . [ m a t h . QA ] J u l heorem 1. Let G be a group acting simply transitively on a product of n ≥ trees ∆ , and let P be a cube complex obtained as the quotient of ∆ by theaction of G . Then G induces a Drinfeld-Manin solution R of the Yang-Baxterequation in the following way: edges of P inherit labelings and orientations fromthe action; the set X is taken to be the set of labels on the edges of the cubes;the bijection R is induced by the 2-cells of P — more precisely, if x i x j x k x l is alabel of a square, then R ( x i , x j ) = ( x − l , x − k ) whereas if x i and x j do not appearnext to each other in a 2-cell of C then R ( x i , x j ) = ( x i , x j ) . The group G is a natural invariant of the Drinfeld-Manin solution; we shallcall it the cube complex group . This distinguishes results of the present paperfrom earlier work. Corollary 1.
Let P be the flip map P : X → X , such that P ( x, y ) = ( y, x ) .Then the map Q = P ◦ R is a solution to the quantum Yang-Baxter equation R R R = R R R , where R is the map from the Theorem 1. The connection between the Yang-Baxter equation and the quantum Yang-Baxter equation was established in [5].
Definition 2.
Two solutions ( X, r ) and ( X (cid:48) , r (cid:48) ) are isomorphic if there exists abijective map ν : X → X (cid:48) such that r (cid:48) ( ν ( x ) , ν ( y )) = ( ν ( σx ( y )) , ν ( γy ( x ))) , where r ( x, y ) = ( σx ( y ) , γy ( x )) , for x, y ∈ X . Proposition 1.
The Drinfeld-Manin solutions associated to non-isometric cubecomplexes are not isomorphic.
Corollary 2.
Formula (4) of Section 3.3 gives a lower bound to the number ofDrinfeld-Manin solutions with | X | = 2( m + l + k ) , where m, l, k ≥ . If we consider a vector space V spanned by elements of X , we get a solution ofthe quantum D -state Yang-Baxter equation, where D = | X | ; see, [7, 4, 5]. Inmost applications to theoretical physics, V is a finite-dimensional space over C .The idea of introducing extra structure on a basis of V goes back to [7], where thebasis vectors were indexed by elements of a finite field or ring. We will show thatone may use even more elaborated algebraic and arithmetic structures: namely,algebras coming from Hamiltonian quaternions and more general quaternionalgebras of finite characteristic. Also, the novelty of our approach lies in thecombination of the arithmetic and algebra with the geometry of CAT (0) cubecomplexes and buildings. Details on how to introduce the structure of a Bruhat-Tits building on a CAT(0) cube complex can be found in [15].2n [10], it was shown that a unitary solution of the Yang-Baxter equationplays the role of a universal quantum gate. It was additionally stated there thatit is surprisingly difficult to produce such solutions. However, every Drinfeld-Manin solution generates unitary solutions of the Yang-Baxter equation and,since we are able to produce several infinite families of such solutions, it wouldbe interesting to investigate possible applications in the spirit of [10].In order to understand the connection between knot theory and solvablemodels, Jones proposed a procedure he called baxterization to produce solu-tions of the Yang-Baxter equation from representations of the braid group, [9],which created a whole area connecting quantum physics, representation the-ory and theory of C*-algebras: namely Temperley-Lieb algebras; see for ex-ample, [8] and references in the paper. The connection with the braid groupand Temperley-Lieb algebras puts some additional restrictions on solutions ofthe Yang-Baxter equation. The solutions obtained by “complexification” ofDrinfeld-Manin solutions do not have these restrictions, so we plan to investi-gate if further generalizations of universal quantum gates are possible [10]. Inthe case of CAT(0) complexes covered by product of trees, we may produceanother class of C*-algebras which are closely related to higher-dimensionalgeneralizations of Thompson groups and can be viewed as higher-dimensionalanalogues of Cuntz-Krieger algebras [11, 18].We would like to mention another potential application here. Possible con-nections of quantum computation and geometric group theory/combinatorics ofwords were indicated in [12], but these connections were not fully developed,since quantum mechanics requires explicit constructions of geometric objectsof dimensions three and higher, and these objects have been constructed onlyrecently [15].
The standard invariants of solutions of the Yang-Baxter equation are structuregroups and structure semigroups (see, for example, [5]) which can be definedfor any mappings X → X for any set X . Definition 3.
Let X be a set and S : X → X a mapping. The structure group G ( X ) is the group generated by elements of X with defining relations xy = tz when S ( x, y ) = ( t, z ) . Definition 4.
Let X be a set and S : X → X a mapping. The structuresemigroup G (cid:48) ( X ) is the semigroup generated by elements of X with definingrelations xy = tz when S ( x, y ) = ( t, z ) . The structure semigroups of the Drinfeld-Manin solutions also posses thestructure of higher-rank graphs with one vertex considered, for example, in [11].There is a connection made between the Yang-Baxter equation and higher-rankgraphs in [21], but concrete examples involve a very particular class of solutionsof the Yang-Baxter equation, different from ours (one way to show this is toconstruct the geometric group and its homology groups).3et
R, P be from Theorem 1. We observe that there is an involution δ onthe set X induced by the directions of edges. Then the relations of the structuresemigroup naturally fall into four equivalence classes corresponding to geometricsquares of the complex P . It is easy to see that the complex cube group is aquotient of the structure group. We first introduce square complexes covered by products of two trees and thendescribe how cube complexes covered by products of n trees can be obtainedfrom n square complexes. The general reference for CAT (0) cube complexes is[16].A square complex S is a 2-dimensional combinatorial cell complex: its 1-skeleton consists of a graph S = ( V ( S ) , E ( S )) with set of vertices V ( S ), andset of oriented edges E ( S ). The 2-cells of the square complex come from a setof squares S ( S ) that are combinatorially glued to the graph S as explainedbelow. Reversing the orientation of an edge e ∈ E ( S ) is written as e (cid:55)→ e − andthe set of unoriented edges is the quotient set E ( S ) = E ( S ) / ( e ∼ e − ) . More precisely, a square (cid:3) is described by an equivalence class of 4-tuplesof oriented edges e i ∈ E ( S ) (cid:3) = ( e , e , e , e )where the origin of e i +1 is the terminus of e i (with i modulo 4). Such 4-tuplesdescribe the same square if and only if they belong to the same orbit underthe dihedral action generated by cyclically permuting the edges e i and by thereverse orientation map( e , e , e , e ) (cid:55)→ ( e − , e − , e − , e − ) . We think of a square-shaped 2-cell glued to the (topological realization of the)respective edges of the graph S . For more details on square complexes, werefer the reader to, for example, [2]. Examples of square complexes are givenby products of trees. Remark 1.
We note that in our definition of a square complex, each square isdetermined by its boundary. The group actions considered in the present paperare related only to such complexes.
Let T l denote the l -valent tree. The product of trees M = T m × T l
4s a Euclidean building of rank 2 and a square complex. Here we are interestedin lattices; that is, those groups Γ acting discretely and cocompactly on M respecting the structure of square complexes. The quotient S = Γ \ M is a finitesquare complex, typically with an orbifold structure coming from the stabilizersof cells.We are especially interested in the case where Γ is torsion free and acts simplytransitively on the set of vertices of M . These yield the smallest quotients S without non-trivial orbifold structure. Since M is a CAT(0) space, any finitegroup stabilizes a cell of M by the Bruhat–Tits fixed point lemma. Moreover,the stabilizer of a cell is profinite, hence compact, so that a discrete group Γacts with trivial stabilizers on M if and only if Γ is torsion free.Let S be a square complex. For x ∈ V ( S ), let E ( x ) denote the set oforiented edges originating in x . The link at the vertex x in S is the (undirectedmulti-)graph L k x whose set of vertices is E ( x ) and whose set of edges in L k x joining vertices a, b ∈ E ( x ) are given by corners γ of squares in S containingthe respective edges of S , see [2].A covering space of a square complex admits a natural structure as a squarecomplex such that the covering map preserves the combinatorial structure. Inthis way, a connected square complex admits a universal covering space. Proposition 2.
The universal cover of a finite connected square complex is aproduct of trees if and only if the link L k x at each vertex x is a complete bipartitegraph.Proof. This is well known and follows, for example, from [1, Theorem C]. A vertical/horizontal structure , in short a VH-structure , on a square complex S consists of a bipartite structure E ( S ) = E v (cid:116) E h on the set of unorientededges of S such that for every vertex x ∈ S the link L k x at x is the completebipartite graph on the induced partition of E ( x ) = E ( x ) v (cid:116) E ( x ) h . Edges in E v (resp. in E h ) are called vertical (resp. horizontal) edges. See [20] for generalfacts on VH-structures. The partition size of the VH-structure is the function V ( S ) → N × N on the set of vertices x (cid:55)→ ( E ( x ) v , E ( x ) h )or just the corresponding tuple of integers if the function is constant. Here − )denotes the cardinality of a finite set.We record the following basic fact; see [2] after their Proposition 1.1. Proposition 3.
Let S be a square complex. Then the following are equivalent:1. The universal cover of S is a product of trees T m × T l and the group ofcovering transformations does not interchange the factors.2. There is a VH-structure on S of constant partition size ( m, l ) . (cid:3) orollary 3. Torsion free cocompact lattices Γ in A ut ( T m ) × A ut ( T l ) , not inter-changing the factors and up to conjugation, correspond uniquely to finite squarecomplexes with a VH-structure of partition size ( m, l ) up to isomorphism.Proof. A lattice Γ yields a finite square complex S = Γ \ T m × T l of the desiredtype. Conversely, a finite square complex S with VH-structure of constantpartition size ( m, l ) has universal covering space M = T m × T l by Proposition 3,and the choice of a base point vertex ˜ x ∈ M above the vertex x ∈ S identifies π ( S, x ) with the lattice Γ = A ut ( M/S ) ⊆ A ut ( T m ) × A ut ( T l ). The latticedepends on the chosen base points only up to conjugation.Simply transitive torsion free lattices not interchanging the factors as inCorollary 3 correspond to square complexes with only one vertex and a VH-structure, necessarily of constant partition size. These will be studied in thenext section. -vertex square complexes Let S be a square complex with just one vertex x ∈ S and a VH-structure E ( S ) = E v (cid:116) E h . Passing from the origin to the terminus of an oriented edgeinduces a fixed point free involution on E ( x ) v and on E ( x ) h . Therefore thepartition size is necessarily a tuple of even integers. Definition 5. A vertical/horizontal structure , in short VH-structure , in agroup G is an ordered pair ( A, B ) of finite subsets A, B ⊆ G such that thefollowing holds.1. Taking inverses induces fixed point free involutions on A and B .2. The union A ∪ B generates G .3. The product sets AB and BA have size A · B and AB = BA .4. The sets AB and BA do not contain -torsion.The tuple ( A, B ) is called the valency vector of the VH-structure in G . Similar to what is described in [2, Section 6.1], starting from a VH-structurewe have the following construction(
A, B ) (cid:32) S A,B (1)yields a square complex S A,B with one vertex and VH-structure starting froma VH-structure (
A, B ) in a group G . The vertex set V ( S A,B ) contains just onevertex x . The set of oriented edges of S A,B is the disjoint union E ( S A,B ) = A (cid:116) B with the orientation reversion map given by e (cid:55)→ e − . Since A and B arepreserved under taking inverses, there is a natural vertical/horizontal structure6uch that E ( x ) h = A and E ( x ) v = B . The squares of S A,B are constructed asfollows. Every relation in
G ab = b (cid:48) a (cid:48) (2)with a, a (cid:48) ∈ A and b, b (cid:48) ∈ B (not necessarily distinct) gives rise to a 4-tuple( a, b, a (cid:48)− , b (cid:48)− ) . The following relations are equivalent to (2) a (cid:48) b − = b (cid:48)− a,a − b (cid:48) = ba (cid:48)− ,a (cid:48)− b (cid:48)− = b − a − . and we consider the four 4-tuples so obtained( a, b, a (cid:48)− , b (cid:48)− ) , ( a (cid:48) , b − , a − , b (cid:48) ) , ( a − , b (cid:48) , a (cid:48) , b − ) , ( a (cid:48)− , b (cid:48)− , a, b )as equivalent. A square (cid:3) of S A,B consists of an equivalence class of such4-tuples arising from a relation of the form (2).
Lemma 1.
The link L k x of S A,B in x is the complete bipartite graph L A,B withvertical vertices labelled by A and horizontal vertices labelled by B .Proof. By 3 of Definition 5 every pair ( a, b ) ∈ A × B occurs on the left handside in a relation of the form (2) and therefore the link L k x contains L A,B .If (2) holds, then the set of left hand sides of equivalent relations { ab, a (cid:48) b − , a − b (cid:48) , a (cid:48)− b (cid:48)− } is a set of cardinality 4, because A and B and AB do not contain 2-torsionby Definition 5 1 + 4 and the right hand sides of the equations are unique byDefinition 5 3. Therefore S A,B only contains ( A · B ) / L k x has at most as many edges as L A,B , and, since it contains L A,B , mustagree with it.
Definition 6.
We will call the complex S A,B a ( A, B ) -complex, keepingin mind, that there are many complexes with the same valency vector. Also,if a group is a fundamental group of a ( A, B ) -complex, we will call it a ( A, B ) -group. Example 1.
Figure 1 shows an example of a (4 , -group. a a b b a a b b a a b b a a b b n -Cube groups n -Cube groups We generalise VH-structure to the n -dimensional case. Definition 7. An n -cube structure in a group G is an ordered tuple ( A , . . . , A n ) of finite subsets A i ⊆ G such that the following hold:1. Taking inverses induces fixed point free involutions on A i .2. The union ∪ A i generates G .3. The product sets A i A j and A j A i have size A i · A j and A i A i = A j A i .4. The sets A i A j and A j A i do not contain -torsion.5. Group G acts simply transitively on Cartesian product of n trees.The tuple ( A , . . . , A j ) is called the valency vector of the n cube-structure in G , and sets ( A , . . . , A n ) are called generating sets. We note that each pair A i , A j ⊆ G forms a subgroup M of G equipped withVH-structure. Without loss of generality, we may consider groups with 3-cube structures. Let G be a group with 3-cube structure with generating sets A, B, C ⊆ G . Since G acts simply transitively on a Cartesian product of three trees ∆, the set X = A ∪ B ∪ C labels the edges of the quotient cube complex P . We show thatthe map R from the formulation of the Theorem 1 satisfies the Yang-Baxterequation, namely to show, that for any ( x, y, z ) ∈ X , R R R ( x, y, z ) = R R R ( x, y, z ). Without loss of generality, we have to consider three cases,1. x ∈ A, y ∈ B, z ∈ C ,2. x ∈ A, y ∈ A, z ∈ C ,3. x ∈ A, y ∈ A, z ∈ A . 8ig. 2 a a a a b b b b c c c c We start with the first case. Consider the cube K in P such that xyz is the pathbetween the most distanced vertices in K (path a b c on fig. 2, for example).Then each of R R R and R R R will move xyz to x (cid:48) y (cid:48) z (cid:48) , where x (cid:48) y (cid:48) z (cid:48) isthe most distanced path parallel to xyz in K , (path c b − a − on fig.2). Secondcase follows from the fact that A, C ⊆ G forms a subgroup M of G equippedwith VH-structure. The third case is true, since R is the identity map at A bydefinition. -cube structure In this section we show that the number of non-isomorphic complexes with 3-cube structure grows at least factorially. Every one vertex square complex withVH-structure (
A, B ) leads to a complex with 3-cube structure. Let (
A, B ) be aVH-structure in a group T . We introduce a set C and add to the relations of T all commutators of C with A ∪ B . Then the resulting group G will be equippedwith a 3-cube structure ( A, B, C ). So, the mass formula for the number of onevertex square complexes with VH-structure up to isomorphism applies for thecase of cube complexes.For completeness, we present a mass formula for the number of one ver-tex square complexes with VH-structure up to isomorphism where each squarecomplex is counted with the inverse order of its group of automorphisms as itsweight [17].Let A (resp. B , C ) be a set with fixed point free involution of size 2 m (resp.2 l , 2 k ). In order to count one vertex square complexes S with VH-structurewith vertical/horizontal partition A (cid:116) B of oriented edges we introduce thegeneric matrix X = ( x ab ) a ∈ A,b ∈ B with rows indexed by A and columns indexed by B and with ( a, b )-entry aformal variable x ab . Let X t be the transpose of X , let τ A (resp. τ B ) be thepermutation matrix for e (cid:55)→ e − for A (resp. B ). For a square (cid:3) we set x (cid:3) = (cid:89) e ∈ (cid:3) x e e = ( a, b ) in the link of S originatingfrom (cid:3) and x e = x ab . Then the sum of the x (cid:3) , when (cid:3) ranges over all possiblesquares with edges from A (cid:116) B , reads (cid:88) (cid:3) x (cid:3) = 14 tr (cid:0) ( τ A Xτ B X t (cid:1) ) , and the number of one vertex square complexes S with VH-structure of partitionsize (2 m, n ) and edges labelled by A and B is given by (cid:101) F m,l = 1( ml )! · ∂ ml (cid:81) a,b ∂x ab (cid:18)
14 tr (cid:0) ( τ A Xτ B X t (cid:1) ) (cid:19) ml . (3)Note that this is a constant polynomial.We can turn this into a mass formula for the number of one vertex squarecomplexes with VH-structure up to isomorphism where each square complex iscounted with the inverse order of its group of automorphisms as its weight. Wesimply need to divide by the order of the universal relabelling A ut ( A, τ A ) × A ut ( B, τ B )) = 2 l ( l )! · m ( m )! . Hence the mass of one vertex square complexes with VH-structure is given by F m,l = 12 l + m +2 lm ( l )! · ( m )! · ( ml )! · ∂ ml (cid:81) a,b ∂x ab (cid:16) tr (cid:0) ( τ A Xτ B X t (cid:1) ) (cid:17) ml . (4)The formula (3) reproduces the numerical values of (cid:101) F m,l for small values (2 m, l )that were computed by Rattaggi in [13, Table B.3]. Here small means ml ≤ Several infinite series of groups acting simply-transitively on products of n treeswere constructed in [15] so by Theorem 1 every such group gives a Drinfeld-Manin solution. We adopt results from [15] to give an explicit construction fora map R satisfying Definition 1. Let q be a prime power. We describe thepresentation of Λ S in terms of finite fields only. Let δ ∈ F × q be a generator of the multiplicative group of the field with q elements. If i, j ∈ Z / ( q − Z are i (cid:54)≡ j (mod q − , then 1 + δ j − i (cid:54) = 0, since otherwise1 = ( − q +1 = δ ( j − i )( q +1) (cid:54) = 1 , a contradiction. Then there is a unique x i,j ∈ Z / ( q − Z with δ x i,j = 1 + δ j − i . x i,j we set y i,j := x i,j + i − j , so that δ y i,j = δ x i,j + i − j = (1 + δ j − i ) · δ i − j = 1 + δ i − j . We moreover set l ( i, j ) := i − x i,j ( q − ,k ( i, j ) := j − y i,j ( q − . If α = δ i and β = δ j , then δ k ( i,j ) = δ j − y i,j ( q − = δ j (1+ δ i − j ) − q = δ i + δ j (1 + δ i − j ) q = δ i + δ j ( δ i + δ j ) q · δ jq = ζ α ( β ) β, (5)and δ l ( i,j ) = δ i − x i,j ( q − = δ i (1+ δ j − i ) − q = δ i + δ j (1 + δ j − i ) q = δ i + δ j ( δ i + δ j ) q · δ iq = ζ β ( α ) α. (6)Let now M ⊆ Z / ( q − Z be a union of cosets under ( q − Z / ( q − n .It was shown in [15] that the following groups act simply transitively onproduct of n trees.Γ M,δ = (cid:28) a i for all i ∈ M (cid:12)(cid:12)(cid:12)(cid:12) a i +( q − / a i = 1 for all i ∈ M ,a i a j = a k ( i,j ) a l ( i,j ) for all i, j ∈ M with i (cid:54)≡ j (mod q − (cid:29) if q is odd, and if q is even:Γ M,δ = (cid:28) a i for all i ∈ M (cid:12)(cid:12)(cid:12)(cid:12) a i = 1 for all i ∈ M ,a i a j = a k ( i,j ) a l ( i,j ) for all i, j ∈ M with i (cid:54)≡ j (mod q − (cid:29) . By Theorem 1, the map R is the following: R ( a i , a j ) = ( a k ( i,j ) a l ( i,j ) ) for all i, j ∈ M with i (cid:54)≡ j (mod q −
1) and R ( a i , a j ) = Id otherwise.The structure semigroup of this solution isΓ (cid:48) M,δ = (cid:10) a i for all i ∈ M (cid:12)(cid:12) a i a j = a k ( i,j ) a l ( i,j ) for all i, j ∈ M with i (cid:54)≡ j (mod q − (cid:11) Example 2.
We compute the smallest example in dimension given by q = 5 and M = { i ∈ Z / Z ; 4 (cid:45) i } . The group Γ acts vertex transitively on product of three trees T × T × T , Γ = (cid:42) a , a , a , a , a , a ,b , b , b , b , b , b ,c , c , c , c , c , c (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a i a i +12 = b i b i +12 = c i c i +12 = 1 for all i ,a b a b , a b a b , a b a b , a b a b , a b a b ,a b a b , a b a b , a b a b , a b a b ,a c a c , a c a c , a c a c , a c a c , a c a c ,a c a c , a c a c , a c a c , a c a c ,b c b c , b c b c , b c b c , b c b c , b c b c ,b c b c , b c b c , b c b c , b c b c . (cid:43) . he group Γ induces a Drinfeld-Manin solution R which can be representedby a × matrix. For any set of size k of distinct odd primes, p , . . . , p k , there is a groupacting simply transitively on a product of k trees of valencies p + 1 , . . . , p k + 1,obtained using Hamiltonian quaternion algebras; see [15]. Example 3.
For p = 3 , p = 5 , p = 7 we construct a group G acting simplytransitively on a product of three trees T × T × T , We set a = 1 + j + k, a = 1 + j − k, a = 1 − j − k, a = 1 − j + k,b = 1 + 2 i, b = 1 + 2 j, b = 1 + 2 k, b = 1 − i, b = 1 − j, b = 1 − k,c = 1 + 2 i + j + k, c = 1 − i + j + k, c = 1 + 2 i − j + k, c = 1 + 2 i + j − k,c = 1 − i − j − k, c = 1 + 2 i − j − k, c = 1 − i + j − k, c = 1 − i − j + k. With this notation we have a − i = a i +2 , b − i = b i +3 , and c − i = c i +4 , andusing these abbreviations the following explicit presentation of a group actingsimply transitively and a product of three trees was found in [15]. Γ = (cid:42) a , a b , b , b c , c , c , c (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a b a b , a b a b , a b a b , a b a b , a b a b , a b a b a c a c , a c a c , a c a c , a c a c ,a c a c , a c a c , a c a c , a c a c b c b c , b c b c , b c b c , b c b c , b c b c , b c b c ,b c b c , b c b c , b c b c , b c b c , b c b c , b c b c (cid:43) . The group Γ induces a Drinfeld-Manin solution R which can be representedby a × matrix. To show that R and R are not isomorphic, we compute first homologies ofΓ and Γ . We use MAGMA, but this can be done by direct calculations as well.Since H (Γ ) = Z / Z × Z / Z × Z / Z , H (Γ ) = Z / Z × Z / Z × Z / Z × Z / Z ,then, by Proposition 1, R and R are not isomorphic. References [1] W. Ballmann, M. Brin,
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