Algebra and Discrete Mathematics RESEARCH ARTICLE Volume27(2019). Number 2, pp. 280–291
c
Journal “Algebra and Discrete Mathematics”
Commutator subgroups of the power subgroups
of generalized Hecke groups
Özden Koruoğlu, Taner Meral, and Recep Sahin
Communicated by A. Yu. Olshanskii
A b s t r ac t . Let p, q >2 be relatively prime integers and letHp,q be the generalized Hecke group associated topandq.The
generalized Hecke groupHp,q is generated byX(z) =−(z−λp)−1
and Y(z) = −(z+λq)−1 where λp = 2 cos π
p andλq = 2 cos π q.
In this paper, for positive integer m, we study the commutator subgroups(Hm
p,q)′ of the power subgroupsH m
p,q of generalized Hecke
groupsHp,q. We give an application related with the derived series
for all triangle groups of the form(0;p, q, n),for distinct primesp, qand for positive integern.
1. Introduction
In [12], Hecke introduced the groups H(λ) generated by two linear fractional transformations
T(z) =−1
z and S(z) =−
1
z+λ,
where λ∈R. Hecke showed that H(λ) is discrete if and only ifλ=λq=
2 cos(πq), q > 3 integer, or λ > 2. We consider the former case q > 3
integer and we denote it byHq=H(λq). Hecke groupHq is isomorphic
to the free product of two finite cyclic groups of orders 2and q, i.e., Hq=hT, S :T2=Sq=Ii ∼=C2∗Cq.
2010 MSC:20H10, 11F06.
The first few Hecke groupsHq are H3 = Γ =P SL(2,Z) (the modular group),H4 =H(√2), H5=H(1+
√ 5
2 ),and H6 =H( √
3).It is clear from the above thatHq⊂P SL(2,Z[λq])unlike in the modular group case (the
caseq= 3), the inclusion is strict and the index|P SL(2,Z[λq]) :Hq|is
infinite asHq is discrete whereasP SL(2,Z[λq])is not forq>4.
Lehner and Newman studied in [20] more general classHp,q of Hecke
groupsHq,by taking
X =− 1
z−λp
and Y =− 1
z+λq
,
where p and q are integers such that2 6p6q, p+q >4. The groups Hp,q have the presentation,
Hp,q =hX, Y :Xp =Yq =Ii ∼=Cp∗Cq. (1.1)
We call these groupsgeneralized Hecke groups Hp,q. Generalized Hecke
groups admit representations as triangle Fuchsian groups with one cusp at infinity. More precisely, a triangle group with signature (0;p, q,∞)
is isomorphic to Hp,q. There is a relationship to Veech groups, see [13]
and [37].
We know from [19] that H2,q =Hq,|Hq :Hq,q|= 2, and there is no
groupH2,2. Also, all Hecke groups Hq are included in generalized Hecke
groupsHp,q. Generalized Hecke groupsHp,q and(p, q,∞)−triangle groups
have been also studied by many authors, in [3], [7], [8], [10], [11], [14], [16], [21], [22], [26], [36] and [38].
On the other hand, ifmis a positive integer, then the power subgroup Hm
p,q is generated by the m
th
powers of all elements of Hp,q. AsHp,qm are
fully invariant subgroups, they are normal inHp,q.
Ifmandnare positive integers, then from the definition one can easily deduce that
Hp,qmn 6Hp,qm (1.2)
and that
Hp,qmn 6(Hp,qm)n.
The last two inequalities imply that
Hp,qm ·Hp,qn =Hp,q(m,n).
a) If (m, p) =dand (m, q) = 1,then
Hp,qm =hYi ∗ hXY X−1i ∗ · · · ∗ hXd−1Y X−(d−1)i ∗ hXdi, Hp,qm ∼=Zq∗ · · · ∗Zq
| {z }
dtimes
∗Zp/d,
and the signature of Hp,qm is(0; q(d), p/d,∞). b) If (m, p) = 1 and(m, q) =d, then
Hp,qm =hXi ∗ hY XY−1i ∗ · · · ∗ hYd−1XY−(d−1)i ∗ hYdi, Hp,qm ∼=Zp∗ · · · ∗Zp
| {z }
dtimes
∗Zq/d,
and the signature of Hp,qm is(0; p(d), q/d,∞).
c) Hp,q :Hp,q′ = pq and Hp,q′ is a free group of rank (p−1)(q−1)
with basis
[X, Y], [X, Y2], . . . , [X, Yq−1],
[X2, Y], [X2, Y2], . . . , [X2, Yq−1], . . . ,
[Xp−1, Y], [Xp−1, Y2], . . . , [Xp−1, Yq−1]
and of signature pq−p−q−(p,q)+2
2 ;∞(p,q)
. d) If (p, q) = 1,then
H′
p,q=Hp,qp ∩Hp,qq . (1.3)
e) If (p, q) = 1and if m=pqn,n∈Z+, then the subgroups Hp,qm are free groups.
The power subgroupsHqmof the Hecke groupsHqand their commutator
subgroups (Hm
q )′ have been studied by many authors in [1], [2], [5], [6],
[9], [15], [17], [18], [24], [29], [30], [31], [32], [33] and [35].
Also, in [31], Sahin and Koruoğlu gave an application related with the derived series for all triangle groups of the form (0; 2, q, n),whereq is a odd prime and n is a positive integer. Using some results given in [39] and [31], they showed that there is a nice connection between the derived series for all triangle groups of the form (0; 2, q, n) and the signatures of the power subgroups Hqm of the Hecke groups Hq and their commutator
In this paper, our aim is to generalize some results given in [31] and [32] for the Hecke groupsHq,to generalized Hecke groupsHp,q wherep, q>2
are relatively prime integers.Firstly, we obtain the group structures and the signatures of commutator subgroups(Hp,qm)′ of the power subgroups Hm
p,q
of the Hecke groupsHp,q. We achieve this by applying standard techniques
of combinatorial group theory (the Reidemeister-Schreier method and the permutation method). Also, we make some numerical examples for the casep < q. Finally, for positive integern, we give an application related with the derived series for all triangle groups of the form(0;p, q, n),for distinct primesp andq.
2. Commutator subgroups of the power subgroups
of generalized Hecke groups Hp,q
Theorem 1. Let p, q > 2 be relatively prime integers and let m be a positive integer. If m is coprime to one of the them, say q, and let
d= (m, p), then
i) Hp,qm : (Hp,qm)′=qd.p d,
ii) The group (Hp,qm)′ is a free group of rank 1 +qd−1(pq−p−q), iii) The group(Hp,qm)′ is of indexqd−1 in H′
p,q.
Proof. i) From (1.1), letk1=Y, k2 =XY X−1,· · · , kd=Xd−1Y X−(d−1),
kd+1=Xd. Then the quotient groupHp,qm/(Hp,qm)′ is obtained by adding
the relation kikj = kjki to the relations of Hp,qm, for i 6= j and i, j ∈
{1,2,· · ·, d+ 1}. Thus we have Hp,qm/(Hp,qm)′∼=Z
q×Zq× · · · ×Zq
| {z }
dtimes
×Zp/d.
Therefore, we find the index Hp,qm : (Hp,qm)′=qd.pd.
ii) Now we choose a Schreier transversalΣfor(Hp,qm)′. Here,Σconsists of identity elementI; d1×(q−1)elements of the formkiawhere16i6d and16a6q−1; (pd−1)elements of the formkt
d+1, where16t6
p d−1; d
2
×(q −1)2 elements of the form kaikjb where 1 6 i < j 6 d and
1 6 a, b 6 q−1; d1×(q−1)×(pd −1) elements of the form kaiktd+1 where16i6d,16a6q−1and16t6 pd−1; d3×(q−1)3 elements of the form ka
ikbjkcs where 1 6 i < j < s 6 d and 1 6 a, b, c 6 q−1; d
2
×(q−1)2×(p
d−1)elements of the formkiakbjkdt+1where16i < j 6d,
16a, b6q−1and16t6 pd−1; · · · ; (q−1)d×(pd−1)elements of the formka1
Using the Reidemeister-Schreier method and after some calculations, we have the generators of (Hp,qm)′ as the followings.
There are 1× d2×(q−1)2 generators of the form [ka
i, kjb] where
16i < j 6d,16a, b6q−1;and1× d1×(q−1)×(pd−1) generators of the form[kia, ktd+1]where16i6d,16a6q−1 and 16t6 pd−1. There are 2 × d3 ×(q −1)3 generators of the form [kai, kjbkcs] or
[kiakbj, ksc] (for the difference, please see the place of the comma) where
1 6i < j < s6dand 16a, b, c6q−1; and2× d2×(q−1)2(pd−1)
generators of the form [kia, kbjkdt+1]or [kaikbj, kdt+1]where 16 i < j 6 d,
16a, b6q−1and 16t6 pd−1.
If we continue similarly, then we find that there are (d−1)× dd× (q−1)d generators of the form[ka1
1 , k2a2· · ·kadd]or[k a1
1 k2a2,· · ·kdad]or· · ·
or [ka1
1 k2a2· · · , k
ad
d ] where 1 6 i 6 d, 1 6 ai 6 q −1; and (d−1)× d
d−1
×(q−1)d−1(p
d−1)generators of the form[k a1
1 , k2a2· · ·k
ad−1
d−1 ktd+1] or
[ka1
1 ka22,· · ·k
ad−1
d−1 k
t
d+1]or· · · or[k1a1ka22· · ·k
ad−1
d−1 , k
t
d+1]where16i6d−1,
16ai6q−1and16t6 pd−1. Finally, there ared× dd×(q−1)d×(pd−1)
generators of the form [ka1
1 , ka22· · ·k
ad
d kdt+1]or[k1a1ka22,· · ·k
ad
d ktd+1]or . . . or[ka1
1 k2a2· · ·kdad, ktd+1] where16i6d,16ai 6q−1and16t6 pd−1.
In fact, there are totally generators
d X
i=2
(i−1)
d i
(q−1)i+
d X i=1 i d i
(q−1)i(p
d−1) = 1 +q
d−1(pq−p−q).
Notice that the number of the generators can be also seen from [25]. iii) We know that
Hp,q:Hp,qm=d, Hp,q :Hp,q′ =pq and Hp,q: (Hp,qm)′=qdp.
Then we have
H′
p,q : (Hp,qm)′
=qd−1.
Finally, we find the signature of(Hm
p,q)′as(1+(pq−p−q−1)q
d−1
2 ;∞(q d−1)
).
Here we give some examples.
Example 1. 1) Let p = 2,q = 3 and m = 2. As d= 2, k1 = Y, k2 = XY X−1. Then we getH2
2,3 : (H22,3)′
= 9.Here, a Schreier transversalΣ
k1, k2, k12, k22; 4 elements of the form k1k2, k1k22,k21k2,k21k22 for (H22,3)′. As(pd−1) = 0,the number of the generators of(H2
2,3)′ is 2
X
i=2
(i−1)
2
i
2i= 1·1·22= 4.
According to the Reidemeister-Schreier method, we have the generators of(H2
2,3)′ as the following. There are4 generators of the form
[k1, k2], [k21, k2], [k1, k22] and [k12, k22]. Therefore,the group (H2
2,3)′ is a free group of rank4 and of signature
(1;∞(3)).
Notice that these results coincide with the results given in [24] for the modular group.
2) Let p = 6, q = 7 and m = 10. As d = 2, we have k1 = Y, k2 =XY X−1, k3 =X2 and
H610,7: (H610,7)′= 147.We choose a Schreier
transversalΣ = { I, 12 elements of the form ka
i where 1 6 i 6 2 and
1 6a6 6; 2 elements of the form k3t, where 1 6t 62;36 elements of the formk1akb2 where 1 6 a, b6 6; 24 elements of the formkaik3t where
1 6i62, 1 6a66 and1 6t 62; 72 elements of the form ka
1kb2ktd+1 where 1 6 a, b 6 6 and 1 6 t 6 2} for (H610,7)′. The number of the generators of(H10
6,7)′ is 2
X
i=2
(i−1)
2
i
6i+
2
X
i=1 i
2
i
6i·2
= 1·1·62+ 1·2·6·2 + 2·1·62·2 = 204.
Using the Reidemeister-Schreier method, we get204 generators of (H610,7)′ as the following. There are36 generators of the form,
[ka1, k2b], where16a, b66,
24 generators of the form
[kai, kt3], where16i62,16a66 and16t62,
144generators of the form,
Thus, the group (H610,7)′ is a free g roup of rank 204 and of signature
(99;∞(7)).
3) Let p = 2, q = 3 and m = 3. Since d = 3, we get k1 = X, k2 = Y XY−1, k3 = Y2XY−2. Then we have
H23,3 : (H23,3)′ = 8. We
choose a Schreier transversalΣ ={I, k1, k2, k3, k1k2, k1k3, k2k3, k1k2k3}. As(qd−1) = 0,the number of the generators of (H23,3)′ is
3
X
i=2
(i−1)
3
i
1i= 1·3·12+ 2·1·12 = 5.
According to the Reidemeister-Schreier method and after some calculations, we have the generators of (H23,3)′ as the followings
[k1, k2], [k1, k3], [k2, k3], [k1, k2k3] and [k1k2, k3]. Also the signature of (H23,3)′ is (1;∞(4)).
Notice that these results coincide with the result given in [24] for the modular group.
Now we can give the following result.
Corollary 1. Let p, q>2 be relatively prime integers. We have
H′
p,q = (Hp,qp )′(Hp,qq )′.
Proof. For the proof, we will use the results in [24] and [31]. We know that
(Hp,qp )′ EH′
p,q and (Hp,qq )′EHp,q′ .
Then we have the chains
(Hp,qp )′ ⊆(Hp
p,q)′(Hp,qq )′ ⊆Hp,q′ and (Hp,qp )′⊆(Hp,qq )′(Hp,qq )′ ⊆Hp,q′ .
From the first chain that
Hp,q′ : (Hp,qp )′(Hp,qq )′qp−1,
and the second chain that
Hp,q′ : (Hp,qp )′(Hp,qq )′pq−1.
Asp and q are relatively primes, we find that (qp−1, pq−1) = 1 and
Hp,q′ : (Hp,qp )′(Hp,qq )′= 1
Therefore we have
H′
As a consequence,we have
Corollary 2. Let p and q be distinct primes and let m be a positive integer. Then
i) If (m, p) = 1 and (m, q) = 1, then Hp,qm ∼= Hp,q and so (Hp,qm)′ =
H′
p,q.In this case, the series of the signatures of Hp,q, Hp,qm and (Hp,qm)′,
respectively, is
Hp,q(0;p, q,∞)⊇(Hp,qm)′(
pq−p−q+ 1
2 ;∞). (2.1)
ii) If m is coprime to one of p and q, say q, and if p = (m, p),then Hm
p,q ∼= Hp,qp and so (Hp,qm)′ ∼= (Hp,qp )′. In this case, the series of the
signatures ofHp,q, Hp,qm and (Hp,qm)′,respectively, is
Hp,q(0;p, q,∞)⊇Hp,qm(0;q(p),∞)
⊃(Hp,qm)′(1 +(pq−p−q−1)qp−1
2 ;∞
(qp−1)
).
If (m, p) = p and (m, q) = q, then the factor group Hp,q/Hp,qm is an
infinite group. In this case, we can not say much aboutHm
p,q apart from
the fact that they are all free normal subgroups.
Notice that some cases of the results are previously known, due partly to the relation between generalized Hecke groups and torus knot groups (e.g. the Theorem 1.1 in the case(m, p) = (m, q) = 1,as well as Corollary 1.3. i) in a more general case, wherep, q are not necessarily assumed both prime as here, but only assumed coprime). The statements, formcoprime to bothp, q (i.e. cased= 1 in the Theorem 1.1), there is a coincidence
(Hp,qm)′ = H′
p,q, free group of rank (p−1)(q−1), and an isomorphism
Hp,qm ∼=Hp,q,including some of generating systems, are known, and can
be found in [36]. Also, see [23] for the freeness of H′
p,q and [4] for the
commutator subgroups of knot groups.
3. An application to the triangle groups (0;p, q, n)
Now we give an application to triangle groups of the form(0;p, q, n)
wherep andq are distinct primes,andnis a positive integer.
We use a Fuchsian group Γwith signature(0;p, q, n).It is well known thatΓ is isomorphic to one relator quotient group Hp,q/R(X, Y) of the
one extra relatorR(X, Y) = (XY)n=I to the standard presentation of Hp,q. Thus Γhas the following presentation
Γ∼=hX, Y |Xp =Yq = (XY)n=Ii.
Then, the quotient group Γ/Γ′ is the group obtained by adding the relation XY =Y X to the relations ofΓ. ThenΓ/Γ′ has a presentation
Γ/Γ′ ∼=hX, Y |Xp=Yq= (XY)n=I, XY =Y X
i
Let ℓ(Γ) denote the drive length of ΓandΓ⊲Γ′ ⊲Γ′′⊲· · ·⊲Γ(k)⊲· · · is its derived series. From [39], we know that if Γ is any non-perfect co-compact Fuchsian group, then the drived lengthℓ(Γ) ofΓ is bounded by
4.Now we give the following example:
Example 2. i) If (n, p) = 1 and (n, q) = 1, then we get X = Y = I from the relations (XY)n = (XY)pq = I. Then Γ = Γ′ and therefore
Γ = Γ′ = Γ′′=· · ·= Γ(k)=· · ·.Consequently, we have ℓ(Γ)=∞.
ii) If nis coprime to one ofp and q, sayq, and if (n, p) =p,then we have Y = I, since Yn = Yq = I. Hence we get Γ/Γ′ ∼= Z
p. Using the
Reidemeister-Schreier method, the permutation method and the Riemann-Hurwitz formula, we have the derived series of Γas
Γ(0;p, q, pr)⊇Γ′(0;q, q,· · · , q
| {z }
ptimes , r)
⊇Γ′′((p−2)q
(p−1)−pq(p−2)+ 2
2 ;r, r,| {z· · ·, r} q(p−1)times
)
⊇Γ′′′((pq−p−q)q(p−2)rq
(p−1)−1
−q(p−1)rq(p−1)−2) + 2
2 ;−).
where r ∈ Z+. Here, the quotient groups Γ/Γ′, Γ′/Γ′′ and Γ′′/Γ′′′ are isomorphic toZp,Zq×Zq× · · · ×Zq
| {z }
(p−1)times
andZr×Zr× · · · ×Zr
| {z }
q(p−1)−1times
,respectively. Indeed, there are infinitely many automorphism groups covered byΓwhich are residually soluble (Γ′′′ and all the terms following Γ′′′ in the series). Therefore we find ℓ(Γ)=4.
iii) If (n, p) =pand(n, q) =q,then we getXp =Yq=I.Since pand
qare distinct primes, we haveΓ/Γ′ ∼=Z
pq.Using the Reidemeister-Schreier
[X2, Yq−1], . . . , [Xp−1, Y], [Xp−1, Y2], . . . ,[Xp−1, Yq−1]. Here the only element of finite order isz= (XY)pq and its order isn/(pq).Using the permutation method and the Riemann-Hurwitz formula, Γ′ has signature
(pq−p−q+1
2 ;r)for r ∈Z+. Then the second derived groupΓ′′ is of infinite index inΓ.Therefore we can find the following series:
Γ(0;p, q, pqr)⊃Γ′ = Γ(pq−p−q+ 1
2 ;r)⊃Γ
′′ ⊃ · · ·. (3.1) HereΓ′ is a free product of a finite cyclic group and (p−1)(q−1)infinite cyclic groups andΓ′′is a free group. Also the corresponding quotient groups
Γ/Γ′ andΓ′/Γ′′ areZ
pq andZ×Z× · · · ×Z
| {z }
(p−1)(q−1)times
, respectively.Therefore, we findℓ(Γ)=3.
Remark 1. 1) There are similar results between the derived series for all triangle groupsΓ of the form(0;p, q, n) and the series of the signatures of the power subgroups of the generalized Hecke groupsHp,q and their
commutator subgroups. There are similarities between (1.1),(1.2) and
(2.3),(2.1), respectively.Of course, there are some differences in these signatures, since(XY)n=I inΓand (XY)∞=I inH
p,q.
2) In the previous example, if we take p= 2,q >3 prime, then our results coincide with the results given in [31] and if we takep= 3andq>5
prime, then we obtain the derived series of the triangle group(0; 3, q, n). These triangle groups are studied by many authors (for example, please see, [38] and [8]).
References
[1] F. Ates, I.N.Cangül, E.K. Cetinalp, A.S. Cevik and E. G. Karpuz,On Commutator and Power Subgroups of Some Coxeter Groups, Appl. Math. Inf. Sci. 10, No. 2, 777-783 (2016).
[2] F. Ates and A.S. Cevik,Knit products of some groups and their applications,Rend. Semin. Mat. Univ. Padova, 121, 1–11 (2009).
[3] I. I. Bouw and M. Möller, Teichmüller curves, triangle groups, and Lyapunov exponents, Ann. of Math. (2) 172, no. 1, 139–185 (2010).
[4] G. Burde and H. Zieschang,Knots, Second edition. De Gruyter Studies in Mathe-matics, 5. Walter de Gruyter & Co., Berlin, 2003.
[5] I. N. Cangül, R. Sahin, S. Ikikardes and Ö. Koruoğlu,Power subgroups of some Hecke groups. II., Houston J. Math.33, no. 1, 33–42.
[7] K. Calta and T. A. Schmidt,Infinitely many lattice surfaces with special pseudo-Anosov maps, J. Mod. Dyn. 7, No. 2, 239-254 (2013).
[8] K. Calta and T. A. Schmidt,Continued fractions for a class of triangle groups, J. Aust. Math. Soc. 93, No. 1-2, 21-42 (2012).
[9] A.S. Cevik, N.Y. Özgür, R. Sahin,The extended Hecke groups as semi-direct products and related results,Int. J. Appl. Math. Stat., 13: 63-72 (2008).
[10] B. Demir, Ö. Koruoğlu and R. Sahin,Conjugacy Classes of Extended Generalized Hecke Groups,Rev. Un. Mat. Argentina, 57, No. 1, 49–56 (2016).
[11] B. Demir, Ö. Koruoğlu and R. Sahin,On Normal Subgroups of Generalized Hecke Groups,An. Şt. Univ. Ovidius Constanta, 24, No. 2, (2016).
[12] E. Hecke,Über die Bestimmung Dirichletscher Reihen durch ihre Funktionalgle-ichung, Math. Ann.112, 664-699 (1936).
[13] W. P. Hooper,Grid graphs and lattice surfaces, Int. Math. Res. Not. IMRN , no. 12, 2657–2698 (2013).
[14] S. Huang,Generalized Hecke groups and Hecke polygons, Ann. Acad. Sci. Fenn., Math. 24, No.1, 187-214 (1999).
[15] S. Ikikardes, O. Koruoglu and R. Sahin,Power subgroups groups of some Hecke groups,Rocky Mountain Journal of Mathematics, No. 2, (2006).
[16] Ş. Kaymak, B. Demir, Ö. Koruoğlu and R. Sahin, Commutator Subgroups of Generalized Hecke and Extended Generalized Hecke Groups, An. Ştiinţ. Univ. Ovidius Constanţa Ser. Mat. 26 (2018), no. 1, 159–168.
[17] C. L. Lang and M. L. Lang, Arithmetic and geometry of the Hecke groups, J. Algebra 460, 392–417 (2016).
[18] C. L. Lang and M. L. Lang, Identifying normal and congruence subgroups, http://arxiv.org/pdf/1501.00743.pdf.
[19] J. Lehner,Uniqueness of a class of Fuchsian groups,Illinois J. Math. 19, 308–315 (1975).
[20] J. Lehner, and M. Newman,Real two-dimensional representations of the modular group and related groups, Amer. J. Math. 87, 945–954 (1965).
[21] G. J. Martin,The geometry and arithmetic of Kleinian groups. Handbook of group actions,Vol. I, 411–494, Adv. Lect. Math. (ALM), 31, Int. Press, Somerville, MA, 2015.
[22] H. Movasati and K. M. Shokri,Automorphic forms for triangle groups: integrality properties, J. Number Theory 145, 67–78 (2014).
[23] L. P. Neuwirth,A remark on knot groups with a center, Proc. Amer. Math. Soc. 14, 378-379 (1963).
[24] M. Newman and J. R. Smart,Note on a subgroup of the modular group, Proc. Amer. Math. Soc. 14, 102–104 (1963).
[25] J. Nielsen,Kommutatorgruppen für das freie Produkt von zyklischen Gruppen. (Danish), Mat. Tidsskr. B, 49-56 (1948).
[27] R. Sahin and O. Bizim,Some subgroups of the extended Hecke groupsH(λq), Acta Math. Sci., Ser. B, Engl. Ed.23, No.4, 497-502 (2003).
[28] R. Sahin, O. Bizim, and I. N. Cangül,Commutator subgroups of the extended Hecke groups, Czech. Math. J.54, No.1, 253-259 (2004).
[29] R. Sahin, S. Ikikardes,Squares of congruence subgroups of the extended modular group, Miskolc Math. Notes, 14, 1031-1035 (2013).
[30] R. Sahin, S. Ikikardes, and Ö. Koruoğlu,Some normal subgroups of the extended Hecke groups H(λp), Rocky Mountain J. Math.36, no. 3, 1033–1048 (2006). [31] R. Sahin and Ö. Koruoğlu,Commutator Subgroups of the Power Subgroups of
Hecke Groups H(λq),Ramanujan J., 24, no. 2, 151–159 (2011).
[32] R. Sahin and Ö. Koruoğlu,Commutator subgroups of the power subgroups of Hecke groups H(λq) II, C. R. Math. Acad. Sci. Paris 349, no. 3-4, 127–130 (2011). [33] R. Sahin, Ö. Koruoğlu and S. Ikikardes,On the extended Hecke groups H(λ5),
Algebra Colloq., 13, 17-23 (2006).
[34] R. Sahin, T. Meral and Ö. Koruoglu,Power and Free Normal Subgroups of Generalized Hecke Groups,accepted for publication in Asian-European Journal of
Mathematics https://doi.org/10.1142/S1793557120500801.
[35] Z. Sarıgedik, S. Ikikardes and R. Sahin,Power subgroups of the extended Hecke groups, Miskolc Math. Notes 16, no. 1, 483–490 (2015).
[36] V. V. Tsanov,Triangle groups, automorphic forms, and torus knots, Enseign. Math. (2) 59, no. 1-2, 73–113 (2013).
[37] W. A. Veech,Teichmüller curves in moduli space, Eisenstein series and an appli-cation to triangular billiards, Invent. Math. 97, no. 3, 553–583 (1989).
[38] C. C. Ward,Calculation of Fuchsian groups associated to billiards in a rational triangle, Ergodic Theory Dynam. Systems 18, no. 4, 1019–1042 (1998).
[39] R. Zomorrodian,Residual solubility of Fuchsian groups, Illinois J. Math., 51, no. 3, 697–703 (2007).
C o n ta c t i n f o r m at i o n
Ö. Koruoğlu, T. Meral, R. Sahin
Balıkesir University, 10100 Balıkesir, Turkey E-Mail(s): [email protected],
