1,383 research outputs found
Area preserving group actions on surfaces
Suppose G is an almost simple group containing a subgroup isomorphic to the
three-dimensional integer Heisenberg group. For example any finite index
subgroup of SL(3,Z) is such a group. The main result of this paper is that
every action of G on a closed oriented surface by area preserving
diffeomorphisms factors through a finite group.Comment: Published by Geometry and Topology at
http://www.maths.warwick.ac.uk/gt/GTVol7/paper21.abs.htm
Periodic points of Hamiltonian surface diffeomorphisms
The main result of this paper is that every non-trivial Hamiltonian
diffeomorphism of a closed oriented surface of genus at least one has periodic
points of arbitrarily high period. The same result is true for S^2 provided the
diffeomorphism has at least three fixed points. In addition we show that up to
isotopy relative to its fixed point set, every orientation preserving
diffeomorphism F: S --> S of a closed orientable surface has a normal form. If
the fixed point set is finite this is just the Thurston normal form.Comment: Published by Geometry and Topology at
http://www.maths.warwick.ac.uk/gt/GTVol7/paper20.abs.htm
Zero entropy subgroups of mapping class groups
Let be a compact surface with boundary. We are interested in the question
of how a group action on permutes a finite invariant set . More precisely, how the algebraic properties of the induced group of
permutations of a finite invariant set affects the dynamical properties of the
group. Our main result shows that in many circumstances if the induced
permutation group is not solvable then among the homeomorphisms in the group
there must be one with a pseudo-Anosov component. We formulate this in terms of
the mapping class group relative to the finite set and show the stronger result
that in many circumstances (e.g. if ) this mapping
class group is itself solvable if it has no elements with pseudo-Anosov
components
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