On the occasion of the retirement of Professor Ruth Kellerhals,
we will have a mini-workshop titled ``Growth 5".
Main topics are geometric group theory, hyperbolic geometry and low dimensional topology.
Date: From 10;30 on January 10th to 16:00 on 11th January 2026
Venue：The building No.14 at Waseda campus
Organizers: Yohei Komori (Waseda), Eiko Kin (Osaka), Tomoshige Yukita (Ashikaga)
Eiko Kin (Osaka University)
Title: Braids, stretch factors and periodic solutions of the planar N-body problem
Abstract: We consider collision-free periodic motions of N points in the plane.
When time is taken as a third axis, the trajectories of these N points form an N-braid.
In particular, periodic solutions of the planar N-body problem give rise to such N-braids.
A primitive braid arising from the figure-eight solution in the 3-body problem, proved by Chenciner and Montgomery
is the simplest pseudo-Anosov 3-braid.
Which braid types can be realized as periodic solutions of the N-body problem?
If the braid type is pseudo-Anosov, which stretch factors can occur?
I will report recent progress about these questions.
This talk is based on joint work with Yuika Kajihara and Mitsuru Shibayama.
Eriko Hironaka (Florida State University)
Title: Coxeter links, and birational automorphisms of the complex projective plane
Abstract: The study of Lehmer's number has revealed unexpected
connections between objects coming from group actions on inner product
spaces, associated hyperbolic links and pseudo-Anosov mapping classes,
and birational automorphisms of the complex plane. In this talk we survey
some of these connections focussing on examples associated to the E_n
Coxeter systems.
Jun Murakami (Waseda University)
Title: Complexified tetrahedrons and double twist knots
Abstract: Complexified tetrahedrons are introduced as deformations
of the regular ideal octahedron in the hyperbolic space.
We see that the complement of a double twist knot is divided
into two congruent complexified tetrahedrons.
As an application, the volume conjecture for the double twist
knots is proved.
Livio Liechti (University of Fribourg)
Title: Minimal stretch factors of orientation-reversing pseudo-Anosov maps
Abstract: It is a notoriously difficult problem to find the minimal stretch factor among all pseudo-Anosov maps of a given surface. In the classical case of closed orientable surfaces and orientation-preserving pseudo-Anosov maps, this problem is only solved in genus one and two. In this talk, we determine the orientation-reversing minimisers in genus two, three and potentially four. This is based on joint work in progress with P. Dehornoy, E. Lanneau and Q. Perroud.
Ruth Kellerhals (University of Fribourg)
Title: On the growth of hyperbolic Coxeter groups
Abstract: I shall present some results and speculations about growth of hyperbolic Coxeter groups, its arithmetic nature and connection to hyperbolic volume.
Masato Mimura (Tohoku University)
Title: Invariant quasimorphisms
Abstract: Joint work with Morimichi Kawasaki (Hokkaido), Mitsuaki Kimura (Osaka Dental), Shuhei Maruyama (Kanazawa) and Takahiro Matsushita (Shinshu). For a pair $(G,N)$ of a group and a normal subgroup, we can define a notion of $G$-invariant quasimorphisms on $N$. This concept may be regarded as a "quasification" of that of invariant homomorphisms. We will give an invitation to this notion, with describing why this "quasification" is interesting.
Jun Nonaka (Waseda University Junior and Senior High School)
Title: Volumes and arithmeticity of $\pi /3$-equiangular hyperbolic polyhedra
Abstract: A hyperbolic polyhedron is called $\pi /3$-equiangular if all its diherdral angles are equal to $\pi /3$.
Atkinson showed that ideal regular tetrahedron had the smallest volume among all $\pi /3$-equiangular
hyperbolic polyhedra in 2009.
In this talk, we show that ideal regular cube has the second smallest volume and pentagonal prism has
the third smallest volume among $\pi /3$-equiangular polyhedra.
This is joint work with Han Yoshida.
Makoto Sakuma (Osaka Central Advanced Mathematical Institute and Hiroshima University)
Title: Topological Models for Spherical CR Uniformizations
Abstract: A spherical CR uniformization of a 3-manifold M realizes M as O/G,
where G is a discrete subgroup of PU(2,1),
the holomorphic isometry group of the complex hyperbolic plane,
and O is the domain of discontinuity of G in the ideal boundary.
The first example of a closed (real) hyperbolic 3-manifold
admitting such a structure was given by Richard Evan Schwartz,
by using a complex hyperbolic triangle group.
In this talk, I will explain the purely combinatorial aspect
of my joint work in progress with Yohei Komori and John Parker.
This work constitute the first step of Parker's project,
which aims to extend Schwartz’s construction
to a much broader class of complex hyperbolic triangle groups.
I will emphasize the analogy with the combinatorial structure of the Ford domains
of punctured torus Kleininan groups acting on real hyperbolic 3-space
Tomoshige Yukita (Ashikaga University)
Title: On the arithmetic properties of growth rates of 2- and 3-dimensional Coxeter systems
Abstract: In 1980, Cannon studied the growth series of closed surface groups and triangle groups and showed that their growth rates are Salem numbers, a special class of real algebraic integers. Following Cannon’s work, the growth rates of discrete reflection groups in hyperbolic spaces have been studied from an arithmetic viewpoint. In this talk, we focus on 2- and 3-dimensional Coxeter systems and discuss the relationship between the arithmetic properties of their growth rates and the Euler characteristics of their nerves.