ON THE GROWTH OF HYPERBOLIC 3-DIMENSIONAL
GENERALIZED SIMPLEX REFLECTION GROUPSYOHEI KOMORI AND YURIKO UMEMOTO
Abstract. We prove that the growth rates of three-dimensional generalized simplex reﬂection groups, i.e. three-dimensional non-compact
hyperbolic Coxeter groups with four generators are always Perron numbers.

1. Introduction
A convex polyhedron P of ﬁnite volume in the n-dimensional hyperbolic
space Hn is called a Coxeter polyhedron if its dihedral angles are submultiplesof π. Any Coxeter polyhedron is a fundamental domain of the discretegroup Γ generated by the set S consisting of the reﬂections with respects
to its facets. We call (Γ, S) an n-dimensional hyperbolic Coxeter group. In
particular when P is a (generalized) simplex of Hn , (Γ, S) is also called a(generalized) simplex reﬂection group ([8]). In this situation we can deﬁne
the word length ℓS (x) of x ∈ Γ with respect to S by the smallest integern ≥ 0 for which there exist s1 , s2 , · · · , sn ∈ S such that x = s∑1 s2 · · · sn .
∞kThe growth function fS (t) of (Γ, S) is the formal power series
k=0 ak t
where ak is the number of elements g ∈ Γ satisfying ℓS (g) = k. It is known√
that the growth rate of (Γ, S), ω := lim supk→∞ k ak is bigger than 1 ([3])
and less than or equal to the cardinality |S| of S. By means of CauchyHadamard formula, the radius of convergence R of fS (t) is the reciprocalof ω, i.e. 1/|S| ≤ R < 1. In practice the growth function fS (t) which isanalytic on |t| < R extends to a rational function P (t)/Q(t) on C by analytic
continuation where P (t), Q(t) ∈ Z[t] are relatively prime. There are formulas
due to Solomon and Steinberg to calculate the rational function P (t)/Q(t)from the Coxeter diagram of (Γ, S) ([10, 11]. See also [4]).
Theorem 1. (Solomon’s formula)
The growth function fS (t) of an irreducible spherical Coxeter group (Γ, S)∏can be written as fS (t) = ki=1 [mi + 1] where [n] := 1 + t + · · · + tn−1 and{m1 , m2 , · · · , mk } is the set of exponents of (Γ, S).
Theorem 2. (Steinberg’s formula)
Let (Γ, S) be a hyperbolic Coxeter group. Let us denote the Coxeter subgroup2010 Mathematics Subject Classiﬁcation. Primary 20F55, Secondary 20F65.
Key words and phrases. growth function; Coxeter group; Perron number.
1

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YOHEI KOMORI AND YURIKO UMEMOTOof (Γ, S) generated by the subset T ⊆ S by (ΓT , T ), and denote its growth
function by fT (t). Set F = {T ⊆ S : ΓT is ﬁnite }. Then
∑ (−1)|T |1=.
fS (t−1 )fT (t)T ∈FIn this case, t = R is a pole of fS (t). Hence R is a real zero of the denominator Q(t) closest to the origin 0 ∈ C of all zeros of Q(t). Solomon’s formula
implies that P (0) = 1. Hence a0 = 1 means that Q(0) = 1. Therefore ω > 1,
the reciprocal of R, becomes a real algebraic integer whose conjugates havemoduli less than or equal to the modulus of ω. If t = R is the unique zero ofQ(t) with the smallest modulus, then ω > 1 is a real algebraic integer whoseconjugates have moduli less than the modulus of ω: such a real algebraic
integer is called a Perron number.
For two and three-dimensional cocompact hyperbolic Coxeter groups,Cannon-Wagreich and Parry showed that the growth rates are Salem numbers ([1, 7]), where a real algebraic integer τ > 1 is called a Salem number
if τ −1 is an algebraic conjugate of τ and all algebraic conjugates of τ otherthan τ and τ −1 lie on the unit circle. From the deﬁnition, a Salem number
is a Perron number.
Kellerhals and Perren calculated the growth functions of all four-dimensionalcocompact hyperbolic Coxeter groups with at most 6 generators and showedthat ω are not Salem numbers while they checked that ω are Perron numbers
numerically. ([6]).
In the non-compact case, Floyd proved that the growth rates of twodimensional non-compact hyperbolic Coxeter groups are Pisot-Vijayaraghavan
numbers, where a real algebraic integer τ > 1 is called a Pisot-Vijayaraghavan
number if algebraic conjugates of τ other than τ lie in the unit disk ([2]). APisot-Vijayaraghavan number is also a Perron number by deﬁnition.
From these results for low-dimensional cases, Kellerhals and Perren conjectured that the growth rates of hyperbolic Coxeter groups are always
Perron numbers. In the present paper, we go to the next stage: threedimensional non-compact hyperbolic Coxeter groups of ﬁnite covolume. We
will show that the growth rate of a three-dimensional generalized simplex
reﬂection group is a Perron number.
2. Denominators of growth functions
There are exactly 23 three-dimensional generalized simplex reﬂection groups([5, 8]). By means of Steinberg’s formula we can calculate growth functionsof them.
Proposition 1. The denominator polynomials Q(t) of the growth functions
fS (t) = P (t)/Q(t) of the 23 three-dimensional generalized simplex reﬂectiongroups (Γ, S) are as follows:• (t − 1)(3t2 + t − 1)
• (t − 1)(3t3 + t2 + t − 1)

ON THE GROWTH OF HYPERBOLIC 3-DIMENSIONAL GENERALIZED SIMPLEX REFLECTION GROUPS
3• (t − 1)(2t4 + 3t3 + t2 − 1)
• (t − 1)(t5 + t4 + t − 1)
• (t − 1)(2t5 + t4 + t2 + t − 1)
• (t − 1)(3t5 + t4 + t3 + t2 + t − 1)
• (t − 1)(t7 + t6 + t5 + t4 + t3 − 1)
• (t − 1)(t7 + t6 + t5 + t4 − 1)
• (t − 1)(t7 + t6 + 2t5 + 2t4 + t3 + t2 − 1)
• (t − 1)(t7 + t6 + 2t5 + t4 + t3 + t − 1)
• (t − 1)(t8 + 2t7 + 2t6 + 3t5 + t4 + t3 − 1)
• (t − 1)(t9 + t7 + t6 + t4 + t2 + t − 1)
• (t − 1)(t13 + t12 + 2t11 + 2t10 + 2t9 + 2t8 + 2t7 + 2t6 + 2t5 + t4 + t3 − 1)
• (t − 1)(t2 + t + 1)(t2 + t − 1)
• (t − 1)(t4 + t3 + t2 + t + 1)(t2 + t − 1)
• (t − 1)(t3 + t − 1)
• (t − 1)(t4 + t3 + t2 + t + 1)(t3 + t − 1)
• (t − 1)(t4 + t3 + t2 + t − 1)
• (t − 1)(t4 + t3 + t2 + t + 1)(t4 + t3 + t2 + t − 1)
• (t − 1)(t5 + t4 + t2 − 1)
• (t − 1)(t5 + t3 + t − 1)
• (t − 1)(t6 + t5 + t4 + t3 + t2 + t − 1)
• (t − 1)(t10 + t9 + t8 + t7 + t6 + t5 + t4 + t3 + t2 + t − 1)
We remark that the factor (t − 1) appears in every denominator of fS (t)because of the fact that 1/fS (1) = χ(Γ) = 0 in the odd-dimensional casedue to a result of Serre ([9]).
3. Main result
Theorem 3. The growth rate of a three-dimensional generalized simplex
reﬂection group is a Perron number.
In Table 1 below, we show the distributions of poles of fS (t) for a particular case of three-dimensional generalized simplex reﬂection groups.
By Proposition 1, the following lemma is suﬃcient to prove the theorem.
Lemma 1. Consider the polynomial of degree n ≥ 2g(t) =n∑

ak tk − 1,k=1

where ak is a non-negative integer. We also assume that the greatest common
divisor of {k ∈ N | ak ̸= 0} is 1. Then there is a real number r0 , 0 < r0 < 1which is the unique zero of g(t) having the smallest absolute value of all zerosof g(t).
∑nkProof. Let us put h(t) =k=1 ak t . Note that g(t) = 0 if and only ifh(t) = 1.

4YOHEI KOMORI AND YURIKO UMEMOTO(Step1) Observe h(0) = 0, h(1) > 1, and h(t) is strictly monotone increasing where t is in the open interval (0, 1). From the intermediate value
theorem, there exists the unique real number r0 in (0, 1) such that h(r0 ) = 1.
(Step2) Suppose there exists a complex number z whose absolute value isless than r0 and satisfying the condition h(z) = 1. Denote z = reiθ where0 < r < r0 and 0 6 θ < 2π. Then1 = |h(z)| = |n∑ak (re ) | ≤iθ kk=1n∑|(ak r )ekikθ

|=k=1n∑ak rk = h(r) < h(r0 ) = 1,k=1

which is a contradiction. Hence r0 has the smallest absolute value of all
zeros of g(t).
(Step3) Consider a complex number z whose
value is equal to∑n absoluteiθ
kikθ
r0 . Set z = r0 e and 0 6 θ < 2π. Then 1 = k=1 ak r0 e implies
1=n∑

ak r0k cos kθ ≤k=1n∑

ak r0k = 1

k=1

Hence cos kθ = 1 for any k ∈ N with ak ̸= 0. The assumption that thegreatest common divisor of {k ∈ N | ak ̸= 0} is 1 means that θ = 0.
Therefore z = r0 , and we conclude that r0 is the unique zero of g(t) having
the smallest absolute value of all zeros of g(t).
¤Coxeter diagramfS (t)(t+1)3 (t2 +1)(t2 −t+1)(t2 +t+1)(t−1)(t8 +2t7 +2t6 +3t5 +t4 +t3 −1)
1.0

0.5

poles of fS (t)

-1.5

-1.0

0.5

-0.5

1.0

-0.5

-1.0

Table 1.
4. RemarkBy Proposition 1, the next lemma shows that some growth rates of threedimensional generalized simplex reﬂection groups are not only Perron numbers but also Pisot-Vijayaraghavan numbers (see Table 2 below).

ON THE GROWTH OF HYPERBOLIC 3-DIMENSIONAL GENERALIZED SIMPLEX REFLECTION GROUPS
5Coxeter diagram(t+1)3 (t2 +1)(t2 −t+1)(t−1)(t6 +t5 +t4 +t3 +t2 +t−1)

fS (t)

1.0

0.5

poles of fS (t)

-1.0

0.5

-0.5

1.0

-0.5

-1.0

Table 2.
∑Lemma 2. For n ≥ 2, the polynomial g(t) = nk=1 tk − 1 has the uniquezero in the unit disk {t ∈ C | |t| < 1} and does not have zeros on the unit
circle |t| = 1.
Proof. Deﬁne h1 (t) = tn+1 , h2 (t) = −2t + 1, andh(t) = h1 (t) + h2 (t) = tn+1 − 2t + 1 = (t − 1)g(t).
Then for any 1/2 < r < 1 suﬃciently close to 1, h(r) < 0. Any complex
number t on the circle {t ∈ C | |t| = r} satisﬁes|h1 (t)| = |tn+1 | = rn+1 < 2r − 1 ≤ |2t − 1| = |h2 (t)|.
Because h2 (t) has the unique zero t = 1/2 in the disk |t| < r, it follows from
Rouché’s theorem that h(t) also has the unique zero in the disk |t| < r. Since
this holds for any r < 1 suﬃciently close to 1, it means that h(t), hence g(t)has the unique zero in the unit disk |t| < 1. Finally we show that g(t) does
not have zeros on the unit circle |t| = 1. Suppose there exists θ ∈ R suchthat g(eiθ ) = 0. Then h(eiθ ) = 0 implies that 1 = |ei(n+1)θ | = |2eiθ − 1|.
Hence eiθ = 1, which contradicts to g(1) ̸= 0. Therefore g(t) has the uniquezero in the unit disk {t ∈ C | |t| < 1} and does not have zeros on the unit
circle |t| = 1.
¤
5. acknowledgement
The ﬁrst author was partially supported by Grant-in-Aid for Scientiﬁc Research(C) (19540194), Ministry of Education, Science and Culture of Japan.
The second author was partially supported by the JSPS Institutional Program for Young Researcher Overseas Visits “ Promoting international young
researchers in mathematics and mathematical sciences led by OCAMI ”.
They thank the referee for her (or his) helpful comments.

6YOHEI KOMORI AND YURIKO UMEMOTO

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Osaka City University Advanced Mathematical Institute and Departmentof Mathematics, Osaka City University, 558-8585, Osaka, Japan
E-mail address: komori@sci.osaka-cu.ac.jp
Department of Mathematics, Osaka City University, 558-8585, Osaka, Japan
E-mail address: yuriko.ummt.77@gmail.com