The geometry of the handlebody groups I: Distortion (Q2885381)

Let \(V_g\) be a handlebody of genus \(g\), namely a \(3\)-manifold bounded by a closed orientable surface \(\partial V_g\) of genus \(g\). The handlebody group \(\mathrm{Map}(V_g)\) is the group of isotopy classes of orientation preserving homeomorphisms of \(V_g\). It is identified with the subgroup of the mapping class group \(\mathrm{Map} (\partial V_g)\) defined by isotopy classes of homeomorphisms of \(\partial V_g\) which can be extended to homeomorphisms of \(V_g\) (see [\textit{S. Suzuki}, ``On homeomorphisms of a 3-dimensional handlebody'', Can. J. Math. 29, 111--124 (1977; Zbl 0339.57001)]). In this article, the authors prove that for \(g \geq 2\) the handlebody group \(\mathrm{Map}(V_g)\) is exponentially distorted in the mapping class group \(\mathrm{Map} (\partial V_g)\).NEWLINENEWLINEThe distorsion of a finitely generated subgroup \(H\) of a finitely generated group \(G\) is a measurement of how far the inclusion map is from a quasi-isometric embedding with respect to the word norm. Distortion was already studied for other subgroups of the mapping class group, for example in the case of subsurfaces by the first author [\textit{U. Hamenstädt}, ``Geometry of the mapping class group I: Boundary amenability'', Invent. Math. 175, 545--609 (2009; Zbl 1197.57003)], and in the case of Torelli group in [\textit{N. Broaddus}, \textit{B. Farb} and \textit{A. Putman}, ``Irreducible Sp-representations and subgroup distortion in the mapping class group'', Comment. Math. Helv. 86, 537--556 (2011; Zbl 1295.57021)].NEWLINENEWLINEThe authors first prove the exponential distortion for handlebodies with marked points or spots. To any loop \(\gamma\) on \(\partial V_g\) based at a marked point \(p\), one can associate an element of the relative mapping class group \(\mathrm{Map} (\partial V_g, p)\). This so-called \textit{point-pushing map} is used to exhibit sequences of mapping classes whose word norm grows linearly in \(\mathrm{Map} (\partial V_g)\) but grows exponentially in \(\mathrm{Map} (V_g)\). The same principle applies when one replaces the marked point with a \textit{spot} or with a subsurface. Applying the Birman exact sequence, the authors then prove that \(\mathrm{Map} (V_g)\) is at least exponentially distorted.NEWLINENEWLINETo show the upper bound of the distortion, the authors define and study \textit{disk systems} which are sets of pairwise disjoint non-homotopic essential disks in \(V_g\), and \textit{racks} which are disk systems together with embedded arcs on the boundary \(\partial V_g\) joining the boundaries of the disks. Racks serve as analogs of train tracks for handlebodies, and provide a geometric model of the handlebody group.