Finiteness properties of asymptotically rigid handlebody groups

TL;DR

This work defines asymptotically rigid handlebody groups for tree handlebodies, distinguishing Cantor-end and finite-end constructions. By building a Stein–Farley cube complex and applying Brown’s finiteness criteria, it proves that Cantor-end cases are of type $F_infty$ while the $r$-point end cases are of type $F_{r-1}$ but not $FP_r$. It connects these finiteness properties to Higman–Thompson groups and to stable handlebody homology, showing, in notable cases, that homology agrees with the stable homology of handlebody groups. The analysis hinges on intricate connectivity of descending links across a network of disc- and curve-complexes, using a sequence of join and fiber arguments to verify the hypotheses of Brown’s criterion. Finally, it relates the homology of the asymptotically rigid handlebody groups to the stabilized homology of ordinary handlebody groups via spectral sequence arguments and known stability results.

Abstract

We introduce asymptotically rigid mapping class groups of handlebodies and determine their finiteness properties, which vary depending on the space of ends of the underlying handlebody. As it turns out, in some cases, the homology of these groups coincides with the stable homology of handlebody groups, as studied by Hatcher and Wahl.

Figures (22)

• Figure 1: The handlebody $O_4$, for $O$ a genus 3 handlebody, $d=2$, and $r=3$.
• Figure 2: In blue, a suited handlebody of $\mathcal{T}_{2,4}(O,Y)$, with $O$ a solid torus. In red, a spotted subhandlebody which is not suited.
• Figure 3: A Cantor handlebody (left) and a star handlebody (right).
• Figure 4: A representation of the flow lines.
• Figure 5: A 1-simplex in $\mathcal{P}_d(M_5,A)$.

Theorems & Definitions (72)

• Theorem 1
• Theorem 2
• Corollary 3
• Lemma 1.1
• Definition 1.2
• Definition 1.3
• Definition 1.4
• Remark 1.5
• Theorem 2.1
• Theorem 2.2