## 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 M1, 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 (2006) on distortion elements in diffeomorphism groups of closed surfaces to the case of surfaces with boundary.

Original language | English (US) |
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Pages (from-to) | 203-212 |

Number of pages | 10 |

Journal | Proceedings of the American Mathematical Society |

Volume | 143 |

Issue number | 1 |

State | Published - Jan 1 2015 |

## Keywords

- Distortion element
- Gluing
- Group action
- Heisenberg group
- Manifold with boundary
- Smoothing

## ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics