  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.
