Smooth gluing of group actions and applications

TL;DR

This paper develops a systematic method to smooth group actions on glued manifolds, enabling new smooth surface actions and extending results on distortion elements to surfaces with boundary.

Contribution

It introduces a technique for smoothing glued group actions, facilitating the construction of new smooth surface actions and extending existing results to manifolds with boundary.

Findings

Constructed smooth group actions on surfaces with boundary.

Extended Franks and Handel's results to surfaces with boundary.

Provided a systematic smoothing method for glued actions.

Abstract

Let $M_1$ and $M_2$ be two $n$-dimensional smooth manifolds with boundary. Suppose we glue $M_1$ and $M_2$ along some boundary components (which are, therefore, diffeomorphic). Call the result $N.$ If we have a group $G$ acting continuously on $M_1,$ and also acting continuously on $M_2,$ such that the actions are compatible on glued boundary components, then we get a continuous action of $G$ on $N$ that stitches the two actions together. However, even if the actions on $M_1$ and $M_2$ are smooth, the action on $N$ probably will not be smooth. We give a systematic way of smoothing out the glued $G$-action. This allows us to construct interesting new examples of smooth group actions on surfaces, and to extend a result of Franks and Handel on distortion elements in diffeomorphism groups of closed surfaces to the case of surfaces with boundary.