Using Hoare's Theorem to find the signature of a subgroup of an NEC group
aa r X i v : . [ m a t h . G R ] A ug USING HOARE’S THEOREM TO FIND THESIGNATURE OF A SUBGROUP OF AN NEC GROUP.
DAVID SINGERMAN AND PAUL WATSON Introduction
There is a standard method, due to the first author, [6] of findingthe signature of a subgroup Λ of a Fuchsian group Γ given the per-mutationn representation of Γ on the Λ -cosets. The structure of anon-Euclidean crystallographic (NEC) group is considerably more com-plicated, but a method of computing the signature of a subgroup wasfound by A. H. M. Hoare in [3] The purpose of this article is to explainhow Hoare’s Theorem is used and to give examples. Partial techniquesto find the signature of a subgroup a are due to Gromadzki, [2].The signature of a cocompact NEC group Γ has the form ( g ; ± ; [ m , . . . m r ]; { ( n , . . . n s ) , . . . { ( n k , . . . n ks k ) } ) (1) Here g is the genus of the quotient space H / Γ , ( H being the hyperbolicplane), m , . . . , m r are the proper periods and the n ij are the link periods; the brackets ( n u . . . n u su ) are called the period cycles. Our problem is to find the signature of a subgroup Λ of finite index in Γ , given the permutation representation of Γ on the Λ cosets.If Γ is a Fuchsian group then the result is in [6]. For Γ a proper NECgroup the result follows from Hoare’s theorem in [3]. This is rathermore complicated to apply than for Fuchsian groups and the purposeof this note is to give a fairly algorithmic procedure to apply Hoare’sresult. (Much of this is derived from results in the second author’sSouthampton Ph.D. thesis,[8].)The proper periods of the subgroup are computed in the same way asfor Fuchsian groups. The major problem is to compute the link periodsand the period cycles for Λ .All reflections of Λ are conjugate to one of the generating reflectionsof Γ and so these correspond to fixed points of the generating reflec-tions of Γ when acting on the right cosets of the subgroup by rightmultiplication. Thus if a reflection generator c i of Γ fixes a coset Λ g j then c ij = g j c i g − j ∈ Λ . We call c ij an induced reflection of Λ . Now
Date : October 29, 2018. suppose that we have a period cycle ( n i , n i . . . n is i ) . This correspondsto a part of the presentation of Γ which has generators c i , . . . c is i , e obeying the relations c ij − = c ij = ( c ij − c ij ) n ij = 1 = e i c e − i c s = 1 (2) We also have elliptic generators x i of orders m i so we have relations x m j j = 1 (3) and also the long relations x . . . x r e . . . e k a b a − b − . . . a g b g a − g b − g = 1 (4) if H / Γ is orientable or x . . . x r e . . . e k a . . . a g = 1 (5) if H / Γ is non-orientable.The group Γ has a presentation which consists of the generators c , . . . , c k , e , . . . , e k , a , b , . . . , a g , b g if H / Γ is orientable and c . . . , c k , e , . . . , e k , a , . . . , a g if H / Γ is non-orientable, and relations (2), (3),(4) if H / Γ is orientableand (2),(3), (5) if H / Γ is non-orientable.The elements c i are reflections, x i elliptics, a i , b i hyperbolic in the ori-entable case and the a i are glide-reflections in the non-orientable case.The e i , called the connecting generators, are orientation-preserving andnearly always hyperbolic.For these presentations see [5], [1]. Definition.
Two reflections c and d of Γ are called linked with linkperiod n if cd is an elliptic element of order n .The following ideas come from Hoare’s paper [3].If c and d are linked reflection generators with link period n then c and d generate a dihedral group D n of order n . Let σ be an orbitof the Λ -cosets under D n and let K be a coset in σ . If m is the leastpositive integer such that ( cd ) m ∈ K then either;a) σ contains no coset fixed by c or d in which case σ has length m and gives an elliptic generator for Λ with period n/m , orb) σ contains two cosets fixed by c and d , one fixed by each if m isodd, two fixed by one and none by the other if m is even. Now σ haslength m and the refection generators of Λ corresponding to the twofixed cosets are linked with period n/m . Each reflection generator of Λ appears in precisely two of these links, unless it is linked only to itself. SING HOARE’S THEOREM TO FIND THE SIGNATURE OF A SUBGROUP OF AN NEC GROUP.3
This gives the period cycles of Λ each coming from one of the periodcycles of Γ . c) In a) above we find out how to get some proper periods of Λ . Otherproper periods of Λ are just those induced by the proper periods of Γ . These are found in exactly the same way as in [6]. That is if x isan elliptic element of Γ of period n , then we let x act on the right Λ -cosets. by right multiplication. If we have an orbit (i.e. a cycle) oflength m then there is an induced elliptic period equal to n/m in Λ .All elliptic periods of Λ are found in a) or c).d) The orientability of Λ is found by using [4]. If Γ is a group with gen-erators Φ and Λ is a subgroup then the Schreier coset graph H (Γ , Λ , Φ) is the graph with vertices the cosets of Λ in Γ and directed edges ateach vertex for each b ∈ Φ such that b : Λ g −→ Λ gb . If c is a re-flection and if Λ gc = Λ g then the directed edge c : Λ g −→ Λ gc is a reflection loop . Let H be the Schreier graph with the reflection loopsdeleted. We call this the augmented Schreier graph. Each path in H corresponds to a word in Φ and hence to an element of Γ . A path iscalled positive (negative) if it corresponds to a orientation-preserving(orientation- reversing ) element of Γ . We label the edges of this graphwith the generators of the group.Then it was shown in [4] that H / Λ is orientable if and only if all circuitsin H are positive, where H is the hyperbolic plane.Finally, the genus of H / Λ is calculated using the Riemann-Hurwitz for-mula as usual. Usually, it is best to find the index | Γ + : Λ + | where Γ + (resp. Λ + ) is the canonical Fuchsian group of Γ , (resp. Λ ) the indextwo subgroups of orientation-preserving isometries. The signature of Γ + is found from [7]. 2. The algorithm.
We suppose that Γ has a subgroup of index N and let the cosets be Λ g , Λ g , . . . Λ g N , which we represent in the usual way by , , . . . .N .Then Step 1.
Write down the induced reflections.
Step 2.
Find all the links and link periods. Typically, c = c ij − and d = c ij are linked with link period n = n ij . Find the image of cd in thesymmetric group S N and note the cycle lengths in this permutation.In particular the the lengths of the cycles containing fixed points of c or d will give the integers m above. DAVID SINGERMAN AND PAUL WATSON
Step 3.
Let c and d be two linked refections with link period n . Then c and d generate a dihedral group D n . Find the orbits of this D n inits action on the Λ -cosets. Let σ be an orbit and let K be a coset in σ (So K is just one of the integers , , · · · , N .) Consider the cycles of cd . If there is a cycle of length m this just means that K ( cd ) m = K . step 4 . Find the periods and link periods of the subgroup. If there areno cosets fixed by c or d in σ then Hoare’s results in [3] tell us that σ has length m and gives an elliptic period n/m in Λ .If there are fixed cosets then σ contains exactly two cosets fixed by c and d , one fixed by each if m is odd, two fixed by one and none by there other if m is even. Now σ has length m and the induced reflection generators of Λ are linked withlink period n/m . We write c linked to d by c ∼ d . Each reflectiongenerator of Λ occurs in exactly two of these links unless it is linkedonly to itself. This gives the period cycles of Λ , each coming from oneof the period cycles of Γ .Step 5. How to deal with the relation ec e − c s = 1 .Regard h ec e − , c s i ∼ = D . Now the orbits of h ec e − , c s i ∼ = D are thetwo-cycles of c s = ec e − or the one-cycles, (fixed points). If we have a2-cycle ( γ, δ ) then { γ, δ } is an orbit containing no fixed points of c s or ec e − (= c s ) . Now ec e − c s = 1 and so m = 1 and there is an ellipticperiod equal to / , that is no elliptic period at all. If c s = ec e − fixes k then k ( ec e − ) = k so c fixes ke and we have a link c s ∼ c ke with link period 1. Example 1
In our first examples we find all subgroups of index two in an extendedtriangle group. This is a group ∆ generated by three reflections in thesides of a triangle with angles π/n , π/n , π/n . This has signature (0; +; [ ]; { n , n , n } ) , which we usually denote by ( n , n , n ) , whichhas presentation { c , c , c | c = c = c = ( c c ) n = ( c c ) n = ( c c ) n = 1 } . There are 7 epimorphisms θ : ∆ −→ C = { e, t } , ( t = 1 ).(1) θ ( c i ) = t for i = 1 , , . Here the kernel (equal to the sta-biliser of a point) is the canonical Fuchsian triangle group withsignature (0; +; [ n , n , n ]; { } ) usually denoted by [ n , n , n ] .(2) θ ( c ) = θ ( c ) = t = (1 , , θ ( c ) = e = (1)(2) . Note that as θ ( c c ) = θ ( c c ) = t , n and n must be even. SING HOARE’S THEOREM TO FIND THE SIGNATURE OF A SUBGROUP OF AN NEC GROUP.5
We now use our algorithm to find Λ the stabiliser of a point.Step 1. The induced reflections are c , c .Step 2. Find the links and link periods. Here the links are c ∼ c with link period n , c ∼ c with link period n and c ∼ c with linkperiod n .Step 3. Find the orbits of the dihedral subgroups. There are threedihedral subgroups here h c , c i ∼ = D l , h c , c i ∼ = D m , and h c , c i ∼ = D n . The orbits of all these three are { , } . Also c c = (1)(2) , and c c = c c = (1 , .Step 4. We first notice that no coset is fixed by c , c . The orbit σ of h c , c i has length 2 so that here m = 1 and so there is an ellipticperiod n in Λ .Now we consider the dihedral group h c , c i . Here the orbit has length2 (which is even) and c fixes two cosets 1 and 2 and c , fixes no cosetso c ∼ c . As c c = (1 , , m = 2 and so the link period is n / .Similarly , we also get c ∼ c from the orbit h c , c i . Here the linkperiod is n / . We get one chain c ∼ c ∼ c and the period cycleis { n / , n / } . As the augmented Schreier graph has two vertices, itcannot have any negative paths and so Λ has orientable quotient space.Thus the signature of Λ is of the form ( g ; +; [ n ]; ( { n / , n / } ) . Wecompute g using Riemann-Hurwitz. We pass to the canonical Fuch-sian groups. ∆ + has Fuchsian signature (0; ( n , n , n )) and Λ + hasFuchsian signature (2 g + 1 − n , n , n / , n / . As the index of λ + in Γ + is two we apply Riemann-Hurwitz to get g − − /n +1 − /n +1 − /n +1 − /n = 2(1 − /n − /n − /n ) giving g = 0 . Thus the signature of the NEC group Λ is (0; +; [ n ]; { ( n / , n / } ) . (6) θ ( c ) = (1 , , θ ( c ) = θ ( c ) = (1)(2) . Again we let Λ denote thestabiliser of a point.Step 1. The induced reflections are c , c , c , c Step 2. The links are as in the previous example.Step 3. The orbits of h c , c i and h c , c i are { , } . The orbits of h c , c i are { } and { } . Also c c = (1 , Step 4. The orbit of h c , c i contains two fixed points of c and noneof c . As c c = (1 , we find a link c ∼ c with link period n / . DAVID SINGERMAN AND PAUL WATSON
Now c c = (1)(2) . By considering the two singleton orbits of h c , c i , we find that the orbit { } contains 1 fixed point of c and one fixedpoint of c . Here c c is the identity so m = 1 (odd!) so we get a link c ∼ c with link period n . Similarly, by considering the orbit { } we find a link c ∼ c with link period n .Also, the orbit of h c , c i contains two fixed points of c and none of c . (Note c c = (1 , so m = 2 , (even!)). Thus c ∼ c with linkperiod n / . We thus have one chain c ∼ c ∼ c ∼ c ∼ c ∼ c with link periods n / , n , n / , n . Thus Λ contains one period cycle ( n / , n , n / , n ) .As in the previous example, we show that we show that Λ has genuszero and orientable quotient space and so Λ has signature (0; +; { ( n / , n , n / , n } ) . (7) By permuting n , n , n we find the other possible subgroups of indextwo leading to the extended triangle group (0; +; [ ]; { n , n , n ) } ) , togive Theorem 1.
The NEC triangle group ∆ of signature (0; +; [ ]; { n , n , n ) } ) has the Fuchsian triangle group [ n , n , n ] as a subgroup of index two.If n , n are even then the only other possible subgroups of index twoin (0; +; [ ]; { n , n , n ) } , have signatures (0; +; [ n ]; { ( n / , n / } (0; +; ; [ ]; { ( n / , n , n / , n } ) .Similarly, if n , n are even we get two other NEC subgroups of index2 and if n , n are even we get another two. Altogether this givesus our seven subgroups of index two in the extended triangle group, ( n , n , n ) . We now give other examples which illustrates all of the points used inHoare’s Theorem. The first one illustrates more fully how to find theperiod cycles and deal with empty period cycles.
Example 2
SING HOARE’S THEOREM TO FIND THE SIGNATURE OF A SUBGROUP OF AN NEC GROUP.7
Let Γ be an NEC group of signature (0; +; [ ] , { (2 , , ( ) } ) (8) The canonical presentation of Γ is h c , c , c , d, e , e | c = c = ( c c ) = ( c c ) = d = 1 , e c e − = c , e de − = d, e e = 1 i . Consider the following permutation representation. c (1 , c (1)(2)(3 , c (1 , e (1)(4)(2 , e (1)(4)(2 , d (1)(4)(2 , We now follow our algorithm.Step 1. The induced reflections are c , c , c , c , c , c , d , d . Step 2. The links are c ∼ c with link period 2, c ∼ c with linkperiod 3, e c e − ∼ c with link period , e de − ∼ d with link period1.Step 3. Find the orbits of the dihedral subgroups. These are h c , c i ∼ = D . The orbits here are { , } , { , } , and c c = (1 , , . h c , c i ∼ = D The orbits are { , , } , { } and c c = (1 , , . h e c e − , c i ∼ = D , The orbits are { , } , { } , { } and ec e − c =(1)(2)(3)(4) . h e de − , d i ∼ = D . The orbits are { , } , { } , { } and ede − d = (1)(2)(3)(4) .Step 4. We have c c = (1 , , . In the cycle (1 , we have twofixed points of c and none of c . (Note m = 2 here is even so weshould have two fixed points of one generator and none of the other.)As n = 2 we have a link c ∼ c with link period / On the orbit { , } of h c , c i we similarly get a link c ∼ c withlink period equal to 1. DAVID SINGERMAN AND PAUL WATSON
Now c c = (1 , , . In the cycle (1 , , we have one fixed point 1of c and one fixed point 4 of c so we get a link c ∼ c with linkperiod / .On the cycle { } of h c , c i we have one fixed point of c and one of c , namely 2 in both cases and so we get a link c ∼ c with linkperiod / .Step 5. We now consider the dihedral group h e c e − , c i ∼ = D of ordertwo. We have two fixed points of c , namely 2 and 4. From step 5, weget links c ∼ c e or c ∼ c with link period 1. Similarly we havethe link c ∼ c with link period 1.Putting all these links together we find that we only have one chain: c ∼ c ∼ c ∼ c ∼ c ∼ c ∼ c All the link periods are equal to one except for the link period between c and c which is equal to 3. We can omit the link periods equal toone and so we are left with a single period cycle { (3) } .We now find the number of period cycles induced by the empty periodcycle of Γ . This empty period cycle corresponds to the reflection d ,which gives two induced reflections d and d . This gives us two emptyperiod cycles of Λ We now consider the orientability of Λ . So we consider the augmentedcoset graph as described above. This graph has 4 vertices 1,2,3,4. Wehave an edge from 1 to 2, because of the reflection c , an edge from1 to 2 coming from the reflection c and an edge from 1 to 3 becauseof the reflection d . Thus we have a triangle −→ −→ −→ . This gives an orientation-reversing loop and so Λ has non-orientablequotient space.We now compute the genus of Λ . The group Γ has signature (8). Λ hasa signature of the form ( g ; − [ ]; { (3) , ( ) , ( )) . By [[7] , Γ + has Fuchsiansignature (1; 2 , and Λ + has signature (g+2; 3). Riemann-Hurwitznow gives g + 2) − / . (7 /
6) = 14 / and thus g = 1 and sothe signature of Λ is (1; [ ]; { (3) , ( ) , ( ) } ) .The next example which comes from [8] illustrates more fully how todetermine the proper periods of the subgroup. Example 3.