### 3-manifold invariants and periodicity of homology spheres.

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We show that the action of the mapping class group on bordered Floer homology in the second to extremal spin${}^{c}$-structure is faithful. This paper is designed partly as an introduction to the subject, and much of it should be readable without a background in Floer homology.

We describe all the group morphisms from the group of orientation-preserving homeomorphisms of the circle to the group of homeomorphisms of the annulus or of the torus.

We consider a large class of non compact hyperbolic manifolds $M={\mathbb{H}}^{n}/\Gamma $ with cusps and we prove that the winding process $\left({Y}_{t}\right)$ generated by a closed $1$-form supported on a neighborhood of a cusp $\mathcal{C}$, satisfies a limit theorem, with an asymptotic stable law and a renormalising factor depending only on the rank of the cusp $\mathcal{C}$ and the Poincaré exponent $\delta $ of $\Gamma $. No assumption on the value of $\delta $ is required and this theorem generalises previous results due to Y. Guivarc’h, Y. Le Jan, J. Franchi and N. Enriquez.

The notion of i-bounded geometry generalises simultaneously bounded geometry and the geometry of punctured torus Kleinian groups. We show that the limit set of a surface Kleinian group of i-bounded geometry is locally connected by constructing a natural Cannon-Thurston map.

Given a finitely generated subgroup G of the group of affine transformations acting on the complex line C, we are interested in the quotient Fix( G)/G. The purpose of this note is to establish when this quotient is finite and in this case its cardinality. We give an application to the qualitative study of polynomial planar vector fields at a neighborhood of a nilpotent singular point.

We show that, if the covering involution of a 3-manifold M occurring as the 2-fold branched covering of a knot in the 3-sphere is contained in a finite nonabelian simple group G of diffeomorphisms of M, then M is a homology 3-sphere and G isomorphic to the alternating or dodecahedral group 𝔸₅ ≅ PSL(2,5). An example of such a 3-manifold is the spherical Poincaré sphere. We construct hyperbolic analogues of the Poincaré sphere. We also give examples of hyperbolic ℤ₂-homology 3-spheres with PSL(2,q)-actions,...