Gr ¨obner–Shirshov Bases for Extended Modular,
Extended Hecke, and Picard Groups
*E. G. Karpuz1** and A. S. ¸Cevik2*** 1Karamano ˘glu Mehmetbey University, Turkey
2Sel ¸cuk University, Turkey
Received December 12, 2008
Abstract—In this paper, Gr ¨obner–Shirshov bases (noncommutative) for extended modular, ex-tended Hecke and Picard groups are considered. A new algorithm for obtaining normal forms of elements and hence solving the word problem in these groups is proposed.
DOI: 10.1134/S0001434612110065
Keywords: extended modular group, extended Hecke group, Gr ¨obner–Shirshov bases, word
problem.
1. INTRODUCTION AND PRELIMINARIES
Algorithmic problems such as the word, conjugacy and isomorphism problems have played an important role in group theory since the work of M. Dehn in the early 1900’s. These problems are called
decision problems, they require a “yes” or “no” answer to a specific question. Among these decision
problems, the word problem in groups and semigroups has been studied most widely (see [1]). It is well known that the word problem for finitely presented groups is not solvable in general; that is, given any two words expressed in the generators of the group, there may be no algorithm to decide whether these words represent the same element in this group.
The method of Gr ¨obner–Shirshov bases, which is the main topic of this paper, gives a new algorithm for obtaining normal forms of elements of groups/semigroups. Therefore, we have a new algorithm for solving the word problem in these groups/semigroups. Having this in mind, our aim in this paper is to find Gr ¨obner–Shirshov bases for extended modular, extended Hecke, and Picard groups.
(a) Gr ¨obner–Shirshov basis. The theories of the Gr ¨obner and Gr ¨obner–Shirshov bases were invented independently by Shirshov [2] (for noncommutative and nonassociative algebras), Hironaka [3], and Buchberger [4] (for commutative algebras). The technique of Gr ¨obner–Shirshov bases has proved to be very useful in the study of presentations of associative algebras, Lie algebras, semigroups, groups, and Ω-algebras by considering generators and relations (see, for example, the book [5] by Bokut and Kukin, and the survey papers [6], [7]).
Let X be a set, let X∗ be the set of X-words (monomials), and let < be a monomial well ordering
of X∗(i.e., < is a well ordering that respects left and right multiplications by words). Also, let k be a field,
and let khXi be the free algebra over X and k. For f ∈ khXi, let f be the maximal (leading) monomial of f . Then f g = f g for any f and g. The polynomial f is said to be monic if the coefficient at f in f is equal to 1. Thus, for monic polynomials f and g, and a, b ∈ X∗the Gr ¨obner–Shirshov basis can be
defined as follows:
(I) Let w be a word such that
w = f b = ag with deg(f) + deg(g) > deg(w). ∗The text was submitted by the authors in English.
**E-mail: eylem.guzel@kmu.edu.tr ***E-mail: sinan.cevik@selcuk.edu.tr
Then the polynomial
(f, g)w = f b − ag
is called the intersection composition of f and g with respect to w. (II) Let w = f = agb. Then the polynomial
(f, g)w = f − agb
is called the inclusion composition of f and g with respect to w. In this case, the transformation f 7→ (f, g)w = f − agb
is called the elimination of the leading word (ELW) of g in f .
Here, the first and second compositions given in (I) and (II) are denoted by f ∧ g and f ∨ g, respec-tively, and the word w is called the ambiguity of the composition (f, g)w. In addition a composition
(f, g)wis called trivial modulo (S, w), in other words, (f, g)w ≡ 0 mod (S, w) if
(f, g)w =
X
αiaisibi and aisibi< w,
where si∈ S, ai, bi∈ X∗, and αi∈ k. In particular, if (f, g)w is zero by ELW’s of polynomials from S,
then
(f, g)w≡ 0 mod (S, w).
Definition 1.1. A monic subset S of khXi is called a Gr ¨obner–Shirshov basis (set) if any composition of polynomials from S is trivial modulo S and modulo the ambiguity. In this case, S is also called a
Gr ¨obner–Shirshov basis of the idealId(S) generated by S and of the algebra khX; Si = khXi/ Id(S) generated by X with defining relations S.
The following lemma is due to Shirshov [2]. It is an analog of the Main Buchberger Theorem [4], [8].
Lemma 1.2 ((Composition–Diamond Lemma). Let S ⊂ khXi be a monic set and let < be a
monomial well ordering ofX∗. Then the following conditions are equivalent:
1) S is a Gr ¨obner–Shirshov basis relative to <;
2) if f ∈ Id(S), then f = asb for some s ∈ S and a, b ∈ X∗;
3) Irr(S) = {u; u 6= asb, s ∈ S, a, b ∈ X∗} is a k-base of the algebra khX; Si.
If S is not a Gr ¨obner–Shirshov basis, then one may construct a Gr ¨obner–Shirshov basis R of Id(S) by adding, at each step, a nontrivial composition of previous polynomials (and reducing it by the ELW’s of previous polynomials and dividing by the leading coefficient). This process is called Shirshov
algorithm. In general, Shirshov algorithm is nonterminating.
The reader is referred to [6], [9]–[14], which are important works on Gr ¨obner–Shirshov basis. (b)Extended modular group, extended Hecke group, and Picard group.
In [15], Hecke introduced an infinite class of discrete groups H(λq) of linear fractional
transforma-tions preserving the upper-half line. The Hecke group is the group generated by x(z) = −1
z and u(z) = z + λq, where λq= 2 cos π/q for some integer q ≥ 3. Let
y = xu = − 1 z + λq
. Then H(λq) has the presentation
For q = 3, the resulting Hecke group H(λ3) = M is the modular group PSL(2, Z). By adding the
inversion r(z) = 1/z to the generators of the modular group, we obtain the extended modular group H(λ3) = M; it was defined in [16]. Then the extended Hecke group, denoted by H(λq), was first
defined in [17] by adding the inversion r(z) = 1/z to the generators of H(λq) as in the extended modular
group M. The Hecke group H(λq) is a subgroup of index 2 in H(λq). By [16], we know that the extended
Hecke group H(λq) is isomorphic to D2∗Z2Dq(where Dqis the dihedral group with 2q elements) and has the presentation
PH(λ
q)= hx, y, r; x
2, yq, r2, (xr)2, (yr)2i. (1)
For q = 3, one obtains the extended modular group M. The Hecke groups H(λq), the extended
Hecke groups H(λq), and their normal subgroups have been extensively studied from many points of
view in the literature (see, [18]–[20]). The Hecke group H(λ3), the modular group PSL(2, Z), and its
normal subgroups have been of great interest in many fields of mathematics, for example number theory, automorphic function theory, and group theory. In [21], these groups were considered from another point of view; it was shown that the extended Hecke group H(λq) is the semi-direct product (split extension)
of the Hecke group H(λq) by a cyclic group of order 2. Moreover, by considering the presentation (1),
the authors of [21] gave necessary and sufficient conditions for (1) to be efficient (which is an algebraic property) on the minimal number of generators. (We refer to [22] for the definition and some details concerning efficiency).
The Picard group P is PSL(2, Z[i]), i.e., the group of linear fractional transformations with Gaussian integer coefficients. The group P is a free free product with amalgamation of the following form:
P = G1∗MG2, where G1 ∼= S3∗Z3 A4, G2 ∼= S3∗Z2 D2
(we recall that D2 is the Klein 4-group), and M is the modular group PSL(2, Z). By [23], it is known
that a presentation for P is given by
PP= hx, u, y, r; x3, u2, y3, r2, (xu)2, (xy)2, (ry)2, (ru)2i, where x(z) = i iz + 1, u(z) = − 1 z, y(z) = z + 1 −z , r(z) = i iz. 2. GR ¨OBNER–SHIRSHOV BASES FOR M, H(λq), AND P
In this section, we find Gr ¨obner–Shirshov bases for the extended modular group M, the extended Hecke group H(λq), and the Picard group P. By the way, we should note that since modular and Hecke
groups are free products of two cyclic groups presented by
H(λ3) = M = hx, y; x2 = y3= 1i and H(λq) = hx, y; x2= yq = 1i,
respectively, their Gr ¨obner–Shirshov bases consist precisely of their own relations.
Now we consider monoid presentations of M, H(λq) and P. Then we take X, Y , R, and U to be
inverse of the generators x, y, r, and u respectively. Now, considering M and H(λq), we order the set
{X, Y, R, x, y, r}∗
lexicographically by using
y < r < x < Y < R < X, and then consider the polynomials
(1) x2− 1, (2) y3− 1, (3) r2− 1, (4) xr − rx, (5) ry2− yr, (6) Xx − 1, (7) xX − 1, (8) Y y − 1, (9) yY − 1, (10) Rr − 1, (11) rR − 1
for extended modular group M and the polynomials
(12) x2− 1, (13) yq− 1, (14) r2− 1, (15) xr − rx, (16) ryq−1− yr, (17) Xx − 1, (18) xX − 1, (19) Y y − 1, (20) yY − 1, (21) Rr − 1, (22) rR − 1
for the extended Hecke group H(λq).
For the Picard group P, we order the set {X, Y, U, R, x, y, u, r}∗ lexicographically by using
x < y < u < r < X < Y < U < R and then consider the polynomials
(23) x3− 1, (24) u2− 1, (25) y3− 1, (26) r2− 1, (27) ux2− xu, (28) y2x2− xy, (29) y2r − ry, (30) ru − ur, (31) Xx − 1, (32) xX − 1, (33) U u − 1, (34) uU − 1, (35) Y y − 1, (36) yY − 1, (37) Rr − 1, (38) rR − 1. The main theorem of this paper is the following.
Theorem 2.1. The Gr ¨obner–Shirshov bases for extended modular groups, extended Hecke
groups, and Picard groups consist of the polynomials (1)–(11), (12)–(22), and (23)–(38), respec-tively.
Proof. We need to prove that all the compositions of the polynomials (1)–(11), (12)–(22), and (23)–
(38) are trivial. If we check for the inclusion compositions of these polynomials, we see that there are no compositions of this type. So it remains to check the intersection compositions of polynomials of each group.
(I)Extended modular groups. Let us consider the intersection compositions of the polynomials (1)– (11). The ambiguities are the following:
(1) ∧ (1) : w = x3, (1) ∧ (4) : w = x2r, (1) ∧ (7) : w = x2X, (2) ∧ (2) : w = y4, (2) ∧ (9) : w = y3Y, (3) ∧ (3) : w = r3, (3) ∧ (5) : w = r2y2, (3) ∧ (11) : w = r2R, (4) ∧ (3) : w = xr2, (4) ∧ (5) : w = xry2, (4) ∧ (11) : w = xrR, (5) ∧ (2) : w = ry3, (5) ∧ (9) : w = ry2Y, (6) ∧ (1) : w = Xx2, (6) ∧ (4) : w = Xxr, (6) ∧ (7) : w = XxX, (7) ∧ (6) : w = xXx, (8) ∧ (2) : w = Y y3, (8) ∧ (9) : w = Y yY, (9) ∧ (8) : w = yY y, (10) ∧ (3) : w = Rr2, (10) ∧ (5) : w = Rry2, (10) ∧ (11) : w = RrR, (11) ∧ (10) : w = rRr. These ambiguities are trivial. Let us show this for some of them.
(1) ∧ (4) : w = x2r,
(f, g)w = (x2− 1)r − x(xr − rx) = x2r − r − x2r + xrx
= xrx − r ≡ 0 (by polynomials (1) and (4)). (2) ∧ (9) : w = y3Y,
(f, g)w = (y3− 1)Y − y2(yY − 1) = y3Y − Y − y3Y + y2
= y2− Y ≡ 0 (by polynomials (2) and (9)). (4) ∧ (5) : w = xry2,
(f, g)w = (xr − rx)y2− x(ry2− yr) = xry2− rxy2− xry2+ xyr
= xyr − rxy2≡ 0 (by polynomials (4) and (5)).
(II)Extended Hecke groups. Let us consider the intersection compositions of the polynomials (12)– (22). Then we have the following ambiguities:
(12) ∧ (12) : w = x3, (12) ∧ (15) : w = x2r, (12) ∧ (18) : w = x2X, (13) ∧ (13) : w = yq+1, (13) ∧ (20) : w = yqY, (14) ∧ (14) : w = r3, (14) ∧ (16) : w = r2yq−1, (14) ∧ (22) : w = r2R, (15) ∧ (14) : w = xr2, (15) ∧ (16) : w = xryq−1, (15) ∧ (22) : w = xrR, (16) ∧ (13) : w = ryq, (16) ∧ (20) : w = ryq−1Y, (17) ∧ (12) : w = Xx2, (17) ∧ (15) : w = Xxr, (17) ∧ (18) : w = XxX, (18) ∧ (17) : w = xXx, (19) ∧ (13) : w = Y yq, (19) ∧ (20) : w = Y yY, (20) ∧ (19) : w = yY y, (21) ∧ (14) : w = Rr2, (21) ∧ (16) : w = Rryq−1, (21) ∧ (22) : w = RrR, (22) ∧ (21) : w = rRr. These ambiguities are trivial. Let us show this for two of them.
(16) ∧ (13) : w = ryq,
(f, g)w = (ryq−1− yr)y − r(yq− 1) = ryq− yry − ryq+ r
= r − yry ≡ 0 (by polynomials (13) and (16)). (17) ∧ (18) : w = XxX,
(f, g)w = (Xx − 1)X − X(xX − 1) = XxX − X − XxX + X ≡ 0.
(III)Picard groups. Let us consider the intersection compositions of the polynomials (23)–(38). In this case, we have the following ambiguities:
(23) ∧ (23) : w = x4, (23) ∧ (32) : w = x3X, (24) ∧ (24) : w = u3, (24) ∧ (27) : w = u2x2, (24) ∧ (34) : w = u2U, (25) ∧ (25) : w = y4, (25) ∧ (28) : w = y3x2, (25) ∧ (29) : w = y3r, (25) ∧ (36) : w = y3Y, (26) ∧ (26) : w = r3, (26) ∧ (30) : w = r2u, (26) ∧ (38) : w = r2R, (27) ∧ (23) : w = ux3, (27) ∧ (32) : w = ux2X, (28) ∧ (23) : w = y2x3, (28) ∧ (32) : w = y2x2X, (29) ∧ (26) : w = y2r2, (29) ∧ (30) : w = y2ru, (29) ∧ (38) : w = y2rR, (30) ∧ (24) : w = ru2, (30) ∧ (27) : w = rux2, (30) ∧ (34) : w = ruU, (31) ∧ (23) : w = Xx3, (31) ∧ (32) : w = XxX, (32) ∧ (31) : w = xXx, (33) ∧ (24) : w = U u2, (33) ∧ (27) : w = U ux2, (33) ∧ (34) : w = U uU, (34) ∧ (33) : w = uU u, (35) ∧ (25) : w = Y y3, (35) ∧ (28) : w = Y y2x2, (35) ∧ (29) : w = Y y2r, (35) ∧ (36) : w = Y yY, (36) ∧ (35) : w = yY y, (37) ∧ (26) : w = Rr2, (37) ∧ (30) : w = Rru, (37) ∧ (38) : w = RrR, (38) ∧ (37) : w = rRr.
These ambiguities are trivial. Let us show this for some of them. (24) ∧ (27) : w = u2x2,
(f, g)w = (u2− 1)x2− u(ux2− xu) = u2x2− x2− u2x2+ uxu
= uxu − x2≡ 0 (by polynomials (24) and (27)). (29) ∧ (30) : w = y2ru,
(f, g)w = (y2r − ry)u − y2(ru − ur) = y2ru − ryu − y2ru + y2ur
= y2ur − ryu ≡ 0 (by polynomials (29) and (30)). Hence we have the result.
The word problem for a group G asks the following question.
Let G be a group given by a finite presentation hS; Ri. Is there an algorithm that decides
whether or not a given word is equivalent to the identity inG?
As we noted in the first section, a Gr ¨obner–Shirshov basis of G yields a set of normal forms of elements of G; that is, for each group element, there is a unique word representing it. Hence, by considering normal forms of elements of G, we can solve the word problem for G. Therefore, by Theorem 2.1, we have the following assertions.
Let C(w1), C(w2), and C(w3) be the normal forms of the words w1 ∈ M, w2∈ H(λq), and w3 ∈ P.
Then the following statement is valid.
Corollary 2.2. The wordsC(w1), C(w2), and C(w3) have the forms
C(w1) = yi1rj1xk1yi2rj2xk2· · · yinrjnxkn, 0 ≤ ip≤ 2, 0 ≤ jq, ks≤ 1, 1 ≤ p, q, s ≤ n; C(w2) = yi1rj1xk1yi2rj2xk2· · · yinrjnxkn, 0 ≤ ip≤ q − 1, 0 ≤ jq, ks≤ 1, 1 ≤ p, q, s ≤ n; C(w3) = xi1yj1uk1rl1xi2yj2uk2rl2· · · xinyjnuknrln, 0 ≤ ip, jq≤ 2, 0 ≤ ks, lt≤ 1, 1 ≤ p, q, s, t ≤ n.
Corollary 2.3. The extended modular groups, the extended Hecke groups, and the Picard groups have a solvable word problem.
REFERENCES
1. S. I. Adyan and V. G. Durnev, “Decision problems for groups and semigroups,” Uspekhi Mat. Nauk 55 (2), 3–94 (2000) [Russian Math. Surveys 55 (2), 207–296 (2000)].
2. A. I. Shirshov, “Some algorithmic problems for Lie algebras,” Sibirsk. Mat. Zh. 3, 292–296 (1962).
3. H. Hironaka, “Resolution of singularities of an algebraic variety over a field of characteristic zero: I,” Ann. Math. 79 (1), 109–203 (1964); “Resolution of singularities of an algebraic variety over a field of characteristic zero: I ,” Ann. Math. 79 (2), 205–326 (1964).
4. B. Buchberger, Ein Algorithmus zum Auffinden der Basiselemente der Restklassenringes nach einem
nulldimensionalen Polynomideal, PhD Thesis (Universit ¨at Innsbruck, 1965).
5. L. A. Bokut’ and G. P. Kukin, Algorithmic and Combinatorial Algebra, in Math. Appl. (Kluwer Acad. Publ., Dordrecht, 1994), Vol. 255.
6. L. A. Bokut’ and P. S. Kolesnikov, “Grobner-Shirshov bases: from their inception to the present time,” in
Problems of Theory of Representation of Lie Algebras and Lie Groups. 7, Zap. Nauchn. Sem. St.-Petersburg. Otdel. Mat. Inst. Steklov (POMI) (POMI, St. Petersburg, 2000), Vol. 272, pp. 26–67 [J.
Math. Sci. (New York)]
7. L. A. Bokut and Y. Chen, “Gr ¨obner–Shirshov bases: some new results,” in Advances in Algebra and
Combinatorics (World Sci. Publ., Hackensack, NJ, 2008), pp. 35–56.
8. B. Buchberger, “Ein algorithmisches Kriterium f ¨ur die L ¨osbarkeit eines algebraischen Gleichungssystems,” Aequationes Math. 4 (3), 374–383 (1970).
9. F. Ate ¸s, E. G. Karpuz, C. Kocapinar, and A. S. ¸Cevik, “Gr ¨obner–Shirshov bases of some monoids,” Discrete Math. 311 (12), 1064–1071 (2011).
10. L. A. Bokut, Y. Chen, and X. Zhao, “Gr ¨obner–Shirshov bases for free inverse semigroups,” Int. J. Algebra Comput. 19 (2), 129–143 (2009).
11. L. Bokut and A. Vesnin, “Gr ¨obner–Shirshov bases for some braid groups,” J. Symbolic Comput. 41 (3-4), 357–371 (2006).
12. Y. Chen, W. Chen, and R. Luo, “Word problem for Novikov’s and Boone’s groups via Gr ¨obner–Shirshov bases,” Southeast Asian Bull. Math. 32 (5), 863–877 (2008).
13. Y. Chen, “Gr ¨obner–Shirshov bases for Schreier extentions of groups,” Comm. Algebra 36 (5), 1609–1625 (2008).
14. Y. Chen and C. Zhong, “Gr ¨obner–Shirshov bases for HNN extentions of groups and for the alternating group,” Comm. Algebra 36 (1), 94–103 (2008).
15. E. Hecke, “ ¨Uber die Bestimmung Dirichletscher Reihen durch ihre Funktionalgleichung,” Math. Ann. 112 (1), 664–699 63.0264.03 JFM 62.1207.01 (1936).
16. G. A. Jones and J. S. Thornton, “Automorphisms and congruence subgroups of the extended modular group,” J. London Math. Soc. (2) 34 (2), 26–40 (1986).
17. S. Huang, “Generalized Hecke groups and Hecke polygons,” Ann. Acad. Sci. Fenn. Math. 24 (1), 187–214 (1999).
18. ¨O. Koruoglu, R. ¸Sahin, and S. ˙Ikikarde ¸s, “The normal subgroup structure of the extended Hecke groups,” Bull. Braz. Math. Soc. (N. S.) 38 (1), 51–65 (2007).
19. ¨O. Koruo ˘glu, R. ¸Sahin, S. ˙Ikikarde ¸s, and I. N. Cang ¨ul, “Normal subgroups of Hecke groups H(λ),” Algebr. Represent. Theory 13 (2), 219–230 (2010).
20. R. ¸Sahin, S. ˙Ikikarde ¸s, and ¨O. Koruo ˘glu, “On the power subgroups of the extended modular group Γ,” Turkish J. Math. 28 (2), 143–151 (2004).
21. A. S. ¸Cevik, N. Y. ¨Ozg ¨ur, and R. ¸Sahin, “The extended Hecke groups as semi-direct products and related results,” Int. J. Appl. Math. Stat. 13 (S08), 63–72 (2008).
22. A. S. ¸Cevik, “The efficiency of standard wreath product,” Proc. Edinburgh Math. Soc. (2) 43 (2), 415–423 (2000).
23. A. M. Brunner, “A two-generator presentation for the Picard group,” Proc. Amer. Math. Soc. 115 (1), 45–46 (1992).
