2015/07/07 by Susumu Hirose, Hirose, Susumu, Eiko Kin +1
Computer Science · Mathematics · #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Group Theory (math.GR) #Mathematical Dynamics and Fractals #math.DS #math.GR #math.GT #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.1507.01671
33 pages, 23 figures; minor changes in exposition, and typos corrected
openalex publication_date 2015/07/07 · openalex created_date 2016/06/24 · arxiv created 2017/02/21 · arxiv updated 2017/02/22 · openalex updated_date 2026/07/28
We consider the hyperelliptic handlebody group on a closed surface of genus g. This is the subgroup of the mapping class group on a closed surface of genus g consisting of isotopy classes of homeomorphisms on the surface that commute with some fixed hyperelliptic involution and that extend to homeomorphisms on the handlebody. We prove that the logarithm of the minimal dilatation (i.e, the minimal entropy) of all pseudo-Anosov elements in the hyperelliptic handlebody group of genus g is comparable to 1/g. This means that the asymptotic behavior of the minimal pseudo-Anosov dilatation of the subgroup of genus g in question is the same as that of the ambient mapping class group of genus g. We also determine finite presentations of the hyperelliptic handlebody groups.