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ANNALES
DE LA FACULTÉ
DES SCIENCES
Mathématiques
FLORENT BALACHEFF, ERAN MAKOVER, HUGO PARLIER
Systole growth for finite area hyperbolic surfaces
Tome XXIII, no 1 (2014), p. 175-180.
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Annales de la Faculté des Sciences de Toulouse Vol. XXIII, n◦ 1, 2014
pp. 175-180
Systole growth for finite area hyperbolic surfaces
Florent Balacheff(1), Eran Makover(2), Hugo Parlier(3)
ABSTRACT. — In this note, we observe that the maximum value achieved
by the systole function over all complete finite area hyperbolic surfaces of a
given signature (g, n) is greater than a function that grows logarithmically
in terms of the ratio g/n.
RÉSUMÉ. — Dans cette note, nous observons que le maximum de la fonc-
tion systole sur l’espace des surfaces hyperboliques complètes et d’aire
finie de signature donnée (g, n) est plus grand qu’une fonction qui crôıt
de façon logarithmique en g/n.
1. Introduction
Consider two natural integers g and n such that 2g − 2 + n is positive.
For an orientable complete finite area hyperbolic surface M of signature
(g, n), where g is the genus and n the number of cusps, the systole is defined
as length of the (or a) shortest closed geodesic and denoted by sys(M).
(∗) Reçu le 30/04/2013, accepté le 23/09/2013
(1) Laboratoire Paul Painlevé, Université Lille 1 Lille, France
florent.balacheff@math.univ-lille1.fr
(2) Central Connecticut State University, Department of Mathematics, New
Britain, CT
makovere@ccsu.edu
(3) Department of Mathematics, University of Fribourg, Switzerland
hugo.parlier@gmail.com
Research supported by Swiss National Science Foundation grant number
PP00P2 128557
Article proposé par Jean-Pierre Otal.
– 175 –
Florent Balacheff, Eran Makover, Hugo Parlier
Because there is an upper bound on the systole which only depends on the
topology of the surface, the quantity
sys(g, n) :=
sup{sys(M) |M is a finite area hyperbolic surface of signature (g, n) }
is finite and we are interested in its asymptotic behavior in terms of g and
n. Observe that this supremum is in fact a maximum following Mahler’s
compactness theorem, see [5].
Schmutz Schaller [6, Theorem 14] proved the following upper bound
sys(g, n)  4 arccosh
(
6g − 6 + 3n
n
)
(1.1)
for n  2 and (g, n) = (0, 3) (see [1] for an alternate proof). For closed
surfaces, Buser and Sarnak proved in [3] that there exists a constant U > 0
such that
sys(g, 0)  U log g
for all genus g  2.
In this note, using Buser and Sarnak’s lower bound in the compact case
and an easy packing argument, we prove that Schmutz Schaller’s upper
bound has indeed the correct asymptotic growth.
Theorem. — For any g  2 and n  1,
sys(g, n) > U log
(
g
n+ 1
)
where U denotes Buser and Sarnak’s constant.
This together with inequality (1.1) implies that for any sequence ng
of integers such that limg→∞ g/ng = +∞, the function sys(g, ng) grows
roughly like log (g/ng).
Along the way we also prove the following inequalities.
Proposition. — For all (g, n) such that 2g − 2 + n > 0 and n  2,
sys(g, n) < sys(g, n+ 1).
Acknowledgements. — The authors are grateful to Marc Troyanov for
some interesting discussions and to the referee for useful comments.
– 176 –
Systole growth for finite area hyperbolic surfaces
2. Systole vs. adding cusps
Fix a signature (g, n) with g, n  2 and let M be a hyperbolic closed sur-
face of genus g with n points {p1, . . . , pn} and M ′ be a hyperbolic punctured
surface with n cusps. Suppose that M ′ and M \{p1, . . . , pn} are conformally
equivalent. We denote by Di := D(pi, r) the disks in M of radius r centered
at the pis.
Lemma 2.1. — For any r such that the disks D1, . . . , Dn of radius r are
embedded and pairwise disjoint,
sys(M ′) > min{sys(M), 4r}.
Proof. — A consequence of the Pick-Schwarz lemma – see for instance
[2, Theorem 3.2] – is that the natural map M ′ →M is strictly contracting.
So any closed geodesic of M ′ whose image under the map M ′ → M is not
homotopically trivial has length strictly bounded from below by sys(M). We
will prove that the remaining homotopy classes have geodesic representatives
of length greater than 4r. Among such classes, a class whose geodesic repre-
sentative is of minimal length is necessarily simple as g  2. Consider such
a class in M ′. This class defines a unique homotopy class in M \{p1, . . . , pn}
which in turn is uniquely represented by a minimal piecewise geodesic in M
where the singular points of the curve belong to {p1, . . . , pn}. The number
of these singular points is at least two and so, as two points of the pis are
at distance at least 2r, its length is at least 4r. To conclude the proof, we
again use that the natural map M ′ →M is strictly contracting. 
Using the same argument, we see that if M is a hyperbolic surface of
signature (g, n) with 2g − 2 + n > 0 and admitting an embedded open disk
of radius sys(M)2 centered at a point p, then the unique hyperbolic surface
M ′ of signature (g, n+ 1) and conformally equivalent to M \ {p} satisfies
sys(M ′) > sys(M).
Observe that the above discussion and the following lemma prove the propo-
sition of the introduction.
Lemma 2.2. — On a hyperbolic surface M of genus g with at most two
cusps, there always exists an embedded open disk of radius sys(M)2 .
Proof. — If there is no cusp, this is a straightforward consequence of
the definition of systole.
So suppose that there is at least one cusp on the surface and proceed by
contradiction. Consider a maximal embedded disk D and denote by p its
– 177 –
Florent Balacheff, Eran Makover, Hugo Parlier
center. The existence of such a disk is guaranteed by compactness (the thick
part of the surface is compact) and the fact that there is an upper bound
on the radius of an embedded disk. By assumption, its radius r is strictly
less than sys(M)2 . Due to the maximality of D, its boundary admits at least
one point of self-intersection which we call a self-bumping point of the disk.
Consider the geodesic loop formed by the two radii of the disk joining p to
some self-bumping point. As its length is 2r < sys(M), this bumping loop
must be parallel to a cusp. If there is only one such bumping loop, it is easy
to see that we get a contradiction: by moving our point p in the appropriate
direction, one can increase the radius of our maximal embedded open disk.
So there are at least two bumping loops. Now observe that because the disk
is embedded, two such loops only intersect in p and thus both loops are non
homotopic (see for instance the bigon criterion [4]). So both loops surround
distinct cusps. In particular for the case of a surface with one cusp we have
reached a contradiction. We now suppose for the sequel that M has two
cusps.
Observe that the existence of a third self-bumping point is impossible,
as there are no more cusps to surround. We will get a contradiction in the
event that there are only two self-bumping loops on the maximal embedded
disk by proving that we can move its center in an appropriate direction in
order to increase its radius. Recall that two horocyclic neighborhoods of
area 1 around the cusps are disjoint. Now consider the two geodesic loops
γ1, γ2 based at p and surrounding the two cusps. Consider, for each of the
neighborhoods, the unique shortest geodesic between the neighborhood and
the point p entirely contained in the cylinder bounded by the cusp and the
corresponding geodesic loop (see figure 1).
Figure 1. — A geodesic loop surrounding a cusp
For each loop γk, denote this distance dk and its associated path ck.
Via standard hyperbolic trigonometry considerations, the geometry of this
hyperbolic cylinder between one of the geodesic loops and the associated
cusp is completely determined by either (γk) or dk. More precisely, there
exists a strictly increasing function that associates to the length (γk) the
distance dk. In particular, as (γ1) = (γ2), we deduce that d1 = d2. Now
– 178 –
Systole growth for finite area hyperbolic surfaces
consider the two distance realizing paths c1 and c2 emanating from p. They
locally divide the surface around p into two parts, and as such one can move
the point p in the appropriate part in order to increase its distance to each
of the horocyclic neighborhoods. This leads to a contradiction. 
We now prove our main theorem. It follows easily from the fact that
sys(g, 1) > sys(g, 0) and the following lower bound.
Lemma 2.3. — For all signature (g, n) with g, n  2,
sys(g, n) > min
{
U log g, 2 arccosh
(
2(g − 1)
n
+ 1
)}
where U denotes Buser and Sarnak’s constant.
Proof. — Fix n  2 and a maximal surface M of genus g  2. By [3]
sys(M)  U log g.
Consider a maximal system {Di}Ni=1 of pairwise disjoint disks of radius
r := min
{
U
4
log g,
1
2
arccosh
(
2(g − 1)
n
+ 1
)}
.
Observe that r  sys(M)4 and denote by pi the center of the disk Di. By the
maximality of such a system, M must be covered by the disks {2Di}Ni=1,
where 2Di denotes the disk of center pi and radius 2r. We derive that
N · 2π(cosh 2r − 1)  N ·A(2Di)  A(M) = 4π(g − 1),
so
N  n.
Now let M ′ be the unique surface with n cusps such that M \ {p1, . . . , pn}
and M ′ are conformally equivalent. We conclude by applying Lemma
2.1. 
Remark that in particular we have
sys(g, [gα])  cα log g,
for any α ∈ [0, 1[ with cα := min{U, 2(1− α)}.
– 179 –
Florent Balacheff, Eran Makover, Hugo Parlier
Bibliography
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ana Math. J. 47, no. 2, p. 419-437 (1998).
[2] Beardon (A.), Minda (D.). — The hyperbolic metric and geometric function
theory. Quasiconformal mappings and their applications, p. 9-56, Narosa, New
Delhi (2007).
[3] Buser (P.), Sarnak (P.). — On the period matrix of a Riemann surface of large
genus. With an appendix by J. H. Conway and N. J. A. Sloane. Invent. Math.
117, no. 1, p. 27-56 (1994).
[4] Farb (B.), Margalit (D.). — A primer on mapping class groups. To appear in
Princeton Mathematical Series.
[5] Mumford (D.). — A remark on a Mahler’s compactness theorem. Proc. AMS 28,
no. 1, p. 289-294 (1971).
[6] Schmutz (P.). — Congruence subgroups and maximal Riemann surfaces. J.
Geom. Anal. 4, p. 207-218 (1994).
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