Narrow Results

In this paper, we study the normal generation of the mapping class group. We first show that a mapping class is a normal generator if its restriction on the invariant subsurface normally generates the (pure) mapping class group of the subsurface. As an application, we provided a criterion for reducible mapping classes to normally generate the mapping class groups in terms of its asymptotic translation lengths on Teichmüller spaces. This is an analogue to the work of Lanier–Margalit dealing with pseudo-Anosov normal generators.

In this paper, we study Helly graphs of finite combinatorial dimension, i.e. whose injective hull is finite-dimensional. We describe very simple fine simplicial subdivisions of the injective hull of a Helly graph, following work of Lang. We also give a very explicit simplicial model of the injective hull of a Helly graph, in terms of cliques which are intersections of balls.

We use these subdivisions to prove that any automorphism of a Helly graph with finite combinatorial dimension is either elliptic or hyperbolic. Moreover, every such hyperbolic automorphism has an axis in an appropriate Helly subdivision, and its translation length is rational with uniformly bounded denominator.

We consider the Artin groups of dihedral type I₂(k) defined by the presentation A_k = 〈a,b | prod(a,b;k) = prod(b,a;k)〉 where prod(s,t;k) = ststs …, with k terms in the product on the right-hand side. We prove that the spherical growth series and the geodesic growth series of A_k with respect to the Artin generators {a,b,a^-1, b^-1} are rational. We provide explicit formulas for the series.