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On the growth rate of ideal coxeter groups in hyperbolic 3-space.

1. Introduction. Let P be a hyperbolic Coxeter polytope which is a polytope in hyperbolic space whose dihedral angles are submultiples of n. The set S of reflections with respects to facets of P generates a discrete group [GAMMA] which has P as a fundamental domain. We call ([GAMMA], S) the Coxeter system associated to P. For k [member of] N, let [a.sub.k] be the number of elements of [GAMMA] whose word length with respects to S is equal to k. Then ([GAMMA], S) has the exponential growth rate [tau] = lim [sup.sub.k[right arrow][infinity]] [kth root of [a.sub.k]] which is a real algebraic integer bigger than 1 ([5]). Recently arithmetic properties of the growth rate of hyperbolic Coxeter groups have attracted considerable attention; for two and three-dimensional cocompact hyperbolic Coxeter groups, Cannon Wagreich and Parry showed that their growth rates are Salem numbers ([2,12]), where a real algebraic integer [tau] > 1 is called a Salem number if [[tau].sup.-1] is an algebraic conjugate of t and all algebraic conjugates of [tau] other than t and [[tau].sup.-1] lie on the unit circle. Floyd also proved that the growth rates of two-dimensional cofinite hyperbolic Coxeter groups are Pisot Vijayaraghavan numbers, where a real algebraic integer [tau] > 1 is called a Pisot Vijayaraghavan number if all algebraic conjugates of [tau] other than [tau] lie in the open unit disk ([3]). Kellerhals and Perren conjectured that the growth rates of hyperbolic Coxeter groups are Perron numbers in general, where a real algebraic integer [tau] > 1 is called a Perron number if all algebraic conjugates of [tau] other than [tau] have moduli less than the modulus of [tau] ([9]). Komori and Umemoto proved their conjecture for three-dimensional cofinite hyperbolic Coxeter simplex groups ([10]). In this paper we consider the growth rate of ideal Coxeter groups in hyperbolic 3-space; a Coxeter polytope P is called ideal if all vertices of P are located on the ideal boundary of hyperbolic space. Related to Jakob Steiner's problem on the combinatorial characterization of polytopes inscribed in the two-sphere [S.sup.2], ideal polytopes in hyperbolic 3-space has been studied extensively ([4,13]). We consider the distribution of growth rates of three-dimensional hyperbolic ideal Coxeter groups; the set G of growth rates will be shown to be unbounded above while it has the minimum which is attained by a unique Coxeter group. Kellerhals studied the same problem for two and three-dimensional cofinite hyperbolic Coxeter groups, and Kellerhals and Kolpakov for two and three-dimensional cocompact hyperbolic Coxeter groups ([7,8]). We will also prove that any element of G is a Perron number, which supports the conjecture of Kellerhals and Perren for three-dimensional hyperbolic ideal Coxeter groups. Moreover we will show that any ideal Coxeter group T with n generators has its growth rate t in the closed interval [n - 3, n - 1], and [GAMMA] is right-angled if and only if [tau] = n - 3. We should remark that Nonaka also detected the minimum growth rate of ideal Coxeter groups, and showed all growth rates to be Perron numbers ([11]). Since we used a criterion for growth rates to be Perron numbers (Proposition 1) and a result of Serre (in the proof of Proposition 3), our arguments are shorter that those of Nonaka.

2. Preliminaries. The upper half space [H.sup.3] = {x = ([x.sub.1], [x.sub.2], [x.sub.3]) [member of] [R.sup.3] | [x.sub.3] > 0} with the metric [absolute value of dx]/[x.sub.3] is a model of hyperbolic 3-space, so called the upper half space model. The Euclidean plane [E.sup.2] = {x = ([x.sub.1], [x.sub.2], [x.sub.3]) [member of] [R.sup.3] | [x.sub.3] = 0} and the point at infinity 1 compose the boundary at infinity [partial derivative][H.sup.3] of [H.sup.3]. A subset B [subset] [H.sup.3] is called a hyperplane of [H.sup.3] if it is a Euclidean hemisphere or a half plane orthogonal to [E.sup.2]. When we restrict the hyperbolic metric [absolute value of dx]/[x.sub.3] of [H.sup.3] to B, it becomes a model of hyperbolic plane. We define a polytope as a closed domain P of [H.sup.3] which can be written as the intersection of finitely many closed half spaces [H.sub.B] bounded by hyperplanes B, say P = [intersection] [H.sub.B]. In this presentation of P, [F.sub.B] = P [intersection] B is a hyperbolic polygon of B. [F.sub.B] is called a facet of P, and B is called the supporting hyperplane of [F.sub.B]. If the intersection of two facets [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] of P consists of a geodesic segment, it is called an edge of P; the intersection [intersection] [F.sub.B] of more than two facets is a point, then it is called a vertex of P. If [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] intersect only at a point of the boundary [partial derivative][H.sup.3] of [H.sup.3], it is called an ideal vertex of P. A polytope P is called ideal if all of its vertices are ideal.

A horosphere [SIGMA] of [H.sup.3] based at [upsilon] [member of] [partial derivative][H.sup.3] is defined by a Euclidean sphere in [H.sup.3] tangent to [E.sup.2] at [upsilon] when [upsilon] [member of] [E.sup.2], or a Euclidean plane in [H.sup.3] parallel to [E.sup.2] when [upsilon] = [infinity]. When we restrict the hyperbolic metric of [H.sup.3] to [SIGMA], it becomes a model of Euclidean plane. Let [upsilon] [member of] [partial derivative][H.sup.3] be an ideal vertex of a polytope P in [H.sup.3] and [SIGMA] be a horosphere of [H.sup.3] based at v such that [SIGMA] meets just the facets of P incident to v. Then the vertex link L([upsilon]) := P [intersection] [SIGMA] of [upsilon] in P is a Euclidean convex polygon in the horosphere [SIGMA]. If [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are adjacent facets of P incident to [upsilon], then the Euclidean dihedral angle between [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is equal to the hyperbolic dihedral angle between the supporting hyperplanes [B.sub.1] and [B.sub.2] in [H.sup.3] (cf. [13, Theorem 6.4.5]).

An ideal polytope P is called Coxeter if the dihedral angles of edges of P are submultiples of [pi]. Since any Euclidean Coxeter polygon is a rectangle or a triangle with dihedral angles ([pi]/2, [pi]/3, [pi]/6), ([pi]/2, [pi]/4, [pi]/4) or ([pi]/3, [pi]/3, [pi]/3), we see that the dihedral angles of an ideal Coxeter polytope must be [pi]/2, [pi]/3, [pi]/4 or [pi]/6.

Any Coxeter polytope P is a fundamental domain of the discrete group [GAMMA] generated by the set S consisting of the reflections with respects to its facets. We call ([GAMMA], S) the Coxeter system associated to P. In this situation we can define the word length [l.sub.S](x) of x [member of] [GAMMA] with respect to S by the smallest integer k [greater than or equal to] 0 for which there exist [s.sub.1], [s.sub.2],***, [s.sub.k] [member of] S such that x = [s.sub.1][s.sub.2] *** [s.sub.k]. The growth function [f.sub.S](t) of ([GAMMA], S) is the formal power series [[SIGMA].sup.[infinity].sub.k=0] [a.sub.k][t.sup.k] where [a.sub.k] is the number of elements g [member of] [GAMMA] satisfying [l.sub.S](g) = k. It is known that the growth rate of ([GAMMA], S), [tau] = lim[sup.sub.k[right arrow][infinity]] [k root of [a.sub.k]] is bigger than 1 ([5]) and less than or equal to the cardinality [absolute value of S] of S from the definition. By means of Cauchy-Hadamard formula, the radius of convergence R of [f.sub.S] (t) is the reciprocal of [tau], i.e. 1/[absolute value of S] [less than or equal to] R < l. In practice the growth function fs(t) which is analytic on [absolute value of t] < R extends to a rational function P(t)/Q(t) on C by analytic continuation where P(t),Q(t) [member of] Z[t] are relatively prime. There are formulas due to Solomon and Steinberg to calculate the rational function P(t)/Q(t) from the data of finite Coxeter subgroups of ([GAMMA],S) ([15,16]. See also [6]).

Theorem 1 (Solomon's formula). The growth function [f.sub.S](t) of an irreducible finite Coxeter group (T, S) can be written as [f.sub.S](t) = [[m.sub.1] + 1, [m.sub.2] + 1, ***, [m.sub.k] + 1] where [n] = 1 + t + *** + [t.sup.n-1], [m, n] = [m][n], etc. and {[m.sub.1], [m.sub.2],***, [m.sub.k]} is the set of exponents of ([GAMMA], S).

Theorem 2 (Steinberg's formula). Let ([GAMMA], S) be a hyperbolic Coxeter group. Let us denote the Coxeter subgroup of ([GAMMA], S) generated by the subset T [subset or equal to] S by ([[GAMMA].sub.T], T), and denote its growth function by [f.sub.T](t). Set F = {T [subset or equal to] S : [[GAMMA].sub.T] is finite}. Then

1/[f.sub.S]([t.sup.-1] = [summation over (T[member of]F)] [(-1).sup.[absolute value of T]]/[f.sub.T] (t)

In this case, t = R is a pole of [f.sub.S](t) = P(t)/ Q(t). Hence R is a real zero of the denominator Q(t) closest to the origin 0 [member of] C of all zeros of Q(t). Solomon's formula implies that P(0) = 1. Hence [a.sub.0] = 1 means that Q(0) = 1. Therefore [tau] > 1, the reciprocal of R, becomes a real algebraic integer whose conjugates have moduli less than or equal to the modulus of [tau]. If t = R is the unique zero of Q(t) with the smallest modulus, then [tau] > 1 is a real algebraic integer whose conjugates have moduli less than the modulus of [tau]: such a real algebraic integer is called a Perron number.

The following result is a criterion for growth rates to be Perron numbers.

Proposition 1 ([10], Lemma 1). Consider the following polynomial of degree n [greater than or equal to] 2

g(t) = [n.summation over (k=1)] [a.sub.k][t.sup.k] - 1,

where [a.sub.k] is a non-negative integer. We also assume that the greatest common divisor of {k [member of] N | [a.sub.k] [not equal to] 0} is 1. Then there is a real number [r.sub.0], 0 < [r.sub.0] < 1 which is the unique zero of g(t) having the smallest absolute value of all zeros of g(t).

3. Ideal Coxeter polytopes with 4 or 5 facets in [H.sup.3]. Let p, q, r and s be the number of edges with dihedral angles [pi]/2, [pi]/3, [pi]/4, and [pi]/6 of an ideal Coxeter polytope P in H3. By Andreev theorem [1], we can classify ideal Coxeter polytopes with 4 or 5 facets, and calculate the growth functions [f.sub.S](t) of P by means of Steinberg's formula and also growth rates, see Table I. Every denominator polynomial has a form (t - 1)H(t) and all coefficients of H(t) satisfy the condition of Proposition 1, so that the growth rates of ideal Coxeter polytopes with 4 or 5 facets are Perron numbers.

As an application of the data of Table I, we have the following result.

Proposition 2. The set Q of growth rates of three-dimensional hyperbolic ideal Coxeter polytopes is unbounded above.

Proof. After glueing m copies of the ideal Coxeter pyramid with p = r = 4 along their sides successively, we can construct a hyperbolic ideal Coxeter polytope [P.sub.n] with n = m + 4 facets. In Fig. 1 we are looking at the ideal Coxeter polytope [P.sub.8] with 8 facets from the point at infinity [infinity], which consists of 4 copies of ideal Coxeter pyramid with p = r = 4 whose apexes are located at [infinity]; squares represent bases of pyramids and disks are supporting hyperplanes of these bases. The growth function of [P.sub.n] has the following denominator polynomial

(t - 1)H(t) = (t - 1)(2(n - 3)[t.sup.3] + (n - 4)[t.sup.2] + (n - 3)t - 1),

from which we see that the growth rate of [P.sub.n] diverges when n goes to infinity.

We should remark that all coefficients of H(t) except its constant term are non-negative. Therefore we can apply Proposition 1 to conclude that the growth rate of [P.sub.n] is a Perron number. Moreover H(t) has a unique zero on the unit interval [0,1] and the following inequalities hold:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[FIGURE 1 OMITTED]

They imply that the growth rate of [P.sub.n] satisfies

n - 3 [??] [tau] [??] n - 1,

which will be generalized in the next section.

4. The growth rates of ideal Coxeter polytopes in [H.sup.3]. Recall that p, q, r and s be the number of edges with dihedral angles [pi]/2, [pi]/3, [pi]/4, and [pi]/6 of an ideal Coxeter polytope P in [H.sup.3]. By means of Steinberg's formula, we can calculate the growth function fs (t) of P as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [2, 3] = [2][3], etc. It can be rewritten as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Proposition 3. Put a = p/2, b = q/3, c = r/4, d = s/6. Then

(1) a + b + c + d = n - 2.

Proof. By a result of Serre ([14]. See also [6]) G(1) = [2 ,3 ,4 , 6](1)(2 - n + p/2 + q/3 + r/4 + s/6) = 0

By using this equality (1) we represent H(t) = G(t)/(t - 1) as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

From this formula we have the following result (see also [11], Theorem 3).

Theorem 3. The growth rates of ideal Coxeter polytopes in [H.sup.3] are Perron numbers.

Proof. When n the number of facets satisfies n [??] 6, the equality (1) of Proposition 3 implies a + b + c + d = n [??] 2 = 4. Then all coefficients of H(t) except its constant term are non-negative. Hence Proposition 1 implies the assertion. For n = 4, 5, this claim was already proved in the previous section.

Moreover the equality (1) induces the following two functions [H.sub.1](t) and [H.sub.2](t) satisfying [H.sub.1](t) [??] H(t) [??] [H.sub.2](t) for any t > 0:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Now we assume that n [??] 6. Then all coefficients of [H.sub.1](t) and [H.sub.2](t) except their constant terms are non-negative so that each of them has a unique zero in (0, [infinity]). The following inequalities

[H.sub.1] (1/n-3) = 0, [H.sub.2] (1/n-1) = - 6/[(n-1).sup.5] < 0

guarantee that the zero of H(t) is located in [1/n-1, 1/n-3]. Combining with the similar result for n = 4, 5 in the previous section, we have the following theorem which is our main result.

Theorem 4. The growth rate [tau] of an ideal Coxeter polytope with n facets in [H.sup.3] satisfies

(2) n - 3 [??] [tau] [??] n - 1.

Corollary 1. An ideal Coxeter polytope P with n facets in [H.sup.3] is right-angled if and only if its growth rate r is equal to n - 3.

Proof. The factor H(t) of the denominator polynomial G(t) = (t - 1)H(t) of the growth function of P is equal to H1 (t) if and only if b = c = d = 0, which means that all dihedral angles are [pi]/2.

From the inequality (2), we see that the growth rate [tau] of an ideal Coxeter polytope with n facets with n [??] 6 satisfies [tau] [??] 3. Therefore combining with the result of growth rates for n = 4, 5 shown in the previous section, we also have the following corollary (see also [11], Theorem 4).

Corollary 2. The minimum of the growth rates of three-dimensional hyperbolic ideal Coxeter polytopes is [0.492432.sup.-1] = 2.03074, which is uniquely realized by the ideal Coxeter simplex with p = q = s = 2.

By Yohei KOMORI and Tomoshige YUKITA

Department of Mathematics, School of Education, Waseda University, Nishi-Waseda 1-6-1, Shinjuku-ku, Tokyo 169-8050, Japan

(Communicated by Kenji FUKAYA, M.J.A., Nov. 12, 2015)

Acknowledgements. The authors thank Dr. Jun Nonaka for explaining his paper [11]. They also thank the referee for her (or his) helpful comments.

doi: 10.3792/pjaa.91.155

References

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[2] J. W. Cannon and Ph. Wagreich, Growth functions of surface groups, Math. Ann. 293 (1992), no. 2, 239-257.

[3] W. J. Floyd, Growth of planar Coxeter groups, P.V. numbers, and Salem numbers, Math. Ann. 293 (1992), no. 3, 475-483.

[4] F. Gueritaud, On an elementary proof of Rivin's characterization of convex ideal hyperbolic polyhedra by their dihedral angles, Geom. Dedicata 108 (2004), 111-124.

[5] P. de la Harpe, Groupes de Coxeter infinis non affines, Exposition. Math. 5 (1987), no. 1, 91-96.

[6] G. J. Heckman, The volume of hyperbolic Coxeter polytopes of even dimension, Indag. Math. (N.S.) 6 (1995), no. 2, 189-196.

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[8] R. Kellerhals and A. Kolpakov, The minimal growth rate of cocompact Coxeter groups in hyperbolic 3-space, Canad. J. Math. 66 (2014), no. 2, 354-372.

[9] R. Kellerhals and G. Perren, On the growth of cocompact hyperbolic Coxeter groups, European J. Combin. 32 (2011), no. 8, 1299-1316.

[10] Y. Komori and Y. Umemoto, On the growth of hyperbolic 3-dimensional generalized simplex reflection groups, Proc. Japan Acad. Ser. A Math. Sci. 88 (2012), no. 4, 62-65.

[11] J. Nonaka, The growth rates of ideal Coxeter polyhedra in hyperbolic 3-space, arXiv:1504.06718.

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[16] R. Steinberg, Endomorphisms of linear algebraic groups, Memoirs of the American Mathematical Society, No. 80, Amer. Math. Soc., Providence, RI, 1968.
Table I

(p, q, r, s)    Denominator polynomial

(2, 2, 0, 2)    (t - 1)(3[t.sup.5] + [t.sup.4] + [t.sup.3]
                [t.sup.2] +1 - 1)

(2, 0, 4, 0)    (t - 1)(3[t.sup.3] + [t.sup.2] + t - 1)

(0, 6, 0, 0)    (t - 1)(3[t.sup.2] + t - 1)

(4, 2, 0, 2)    (t - 1)(4[t.sup.5] + [t.sup.4] + 2[t.sup.3] +
                [t.sup.2] + 2t - 1)

(4, 0, 4, 0)    (t - 1)(4[t.sup.3] + [t.sup.2] + 2t - 1)

(2, 5, 0, 2)    (t - 1)(5[t.sup.5] + 2[t.sup.4] + [t.sup.3] +
                3[t.sup.2] + 2t - 1)
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Author:Komori, Yohei; Yukita, Tomoshige
Publication:Japan Academy Proceedings Series A: Mathematical Sciences
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Date:Dec 1, 2015
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