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Smooth gluing of group actions and applications


Author: Kiran Parkhe
Journal: Proc. Amer. Math. Soc. 143 (2015), 203-212
MSC (2010): Primary 37C85; Secondary 57M60, 37C05
DOI: https://doi.org/10.1090/S0002-9939-2014-12231-3
Published electronically: August 22, 2014
MathSciNet review: 3272745
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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 (2006) on distortion elements in diffeomorphism groups of closed surfaces to the case of surfaces with boundary.


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Additional Information

Kiran Parkhe
Affiliation: Department of Mathematics, Technion-Israel Institute of Technology, Haifa 32000, Israel
Email: kiranparkhe2012@u.northwestern.edu

DOI: https://doi.org/10.1090/S0002-9939-2014-12231-3
Keywords: Group action, manifold with boundary, gluing, smoothing, Heisenberg group, distortion element
Received by editor(s): October 30, 2012
Received by editor(s) in revised form: March 14, 2013
Published electronically: August 22, 2014
Communicated by: Nimish Shah
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.