Local connectivity, Kleinian groups and geodesics on the blowup of the torusMcMullen, Curtis T.
doi: 10.1007/PL00005809pmid: N/A
Let N=?3/Γ be a hyperbolic 3-manifold with free fundamental group π1(N)≅Γ≅<A,B>, such that [A,B] is parabolic. We show that the limit set λ of N is always locally connected. More precisely, let Σ be a compact surface of genus 1 with a single boundary component, equipped with the Fuchsian action of π1(Σ) on the circle S
infty
1. We show that for any homotopy equivalence f:Σ?N, there is a natural continuous map¶¶F:S
infty
1?λ⊂S
infty
2,¶¶respecting the action of π1(Σ). In the course of the proof we determine the location of all closed geodesics in N, using a factorization of elements of π1(Σ) into simple loops.
K-area, Hofer metric and geometry of conjugacy classes in Lie groupsEntov, Michael
doi: 10.1007/s002220100161pmid: N/A
Given a closed symplectic manifold (M,ω) we introduce a certain quantity associated to a tuple of conjugacy classes in the universal cover of the group Ham (M,ω) by means of the Hofer metric on Ham (M,ω). We use pseudo-holomorphic curves involved in the definition of the multiplicative structure on the Floer cohomology of a symplectic manifold (M,ω) to estimate this quantity in terms of actions of some periodic orbits of related Hamiltonian flows. As a corollary we get a new way to obtain Agnihotri-Belkale-Woodward inequalities for eigenvalues of products of unitary matrices. As another corollary we get a new proof of the geodesic property (with respect to the Hofer metric) of Hamiltonian flows generated by certain autonomous Hamiltonians. Our main technical tool is K-area defined for Hamiltonian fibrations over a surface with boundary in the spirit of L. Polterovich’s work on Hamiltonian fibrations over S
2.
Bounded geometry for Kleinian groupsMinsky, Yair N.
doi: 10.1007/s002220100163pmid: N/A
We show that a Kleinian surface group, or hyperbolic 3-manifold with a cusp-preserving homotopy-equivalence to a surface, has bounded geometry if and only if there is an upper bound on an associated collection of coefficients that depend only on its end invariants. Bounded geometry is a positive lower bound on the lengths of closed geodesics. When the surface is a once-punctured torus, the coefficients coincide with the continued fraction coefficients associated to the ending laminations.