Accidental Parabolics in Mapping Class Groups
نویسنده
چکیده
In this note we discuss the behavior of the Gromov boundaries and limit sets for the surface subgroups of the mapping class group with accidental parabolics constructed by the author and A. Reid in [8]. Specifically, we show that generically there are no Cannon–Thurston maps from the Gromov boundary to Thurston’s boundary of Teichmüller space.
منابع مشابه
Accidental Parabolics in the Mapping Class Group
In this paper we discuss the behavior of the Gromov boundaries and limit sets for the surface subgroups of the mapping class group with accidental parabolics constructed by the author and A. Reid (2006). Specifically, we show that generically there are no Cannon–Thurston maps from the Gromov boundary to Thurston’s boundary of Teichmüller space.
متن کاملCannon–thurston Maps for Kleinian Groups
We show that Cannon–Thurston maps exist for degenerate free groups without parabolics, that is, for handlebody groups. Combining these techniques with earlier work proving the existence of Cannon–Thurston maps for surface groups, we show that Cannon–Thurston maps exist for arbitrary finitely generated Kleinian groups without parabolics, proving conjectures of Thurston and McMullen. We also show...
متن کاملHausdorff dimension and conformal dynamics I: Strong convergence of Kleinian groups
This paper investigates the behavior of the Hausdorff dimensions of the limit sets Λn and Λ of a sequence of Kleinian groups Γn → Γ, where M = H/Γ is geometrically finite. We show if Γn → Γ strongly, then: (a) Mn = H 3/Γn is geometrically finite for all n ≫ 0, (b) Λn → Λ in the Hausdorff topology, and (c) H. dim(Λn) → H. dim(Λ), if H. dim(Λ) ≥ 1. On the other hand, we give examples showing the ...
متن کاملHausdor dimension and conformal dynamics I: Strong convergence of Kleinian groups
This paper investigates the behavior of the Hausdorr dimensions of the limit sets n and of a sequence of Kleinian groups ? n ! ?, where M = H 3 =? is geometrically nite. We show if ? n ! ? strongly, then: (a) M n = H 3 =? n is geometrically nite for all n 0, (b) n ! in the Hausdorr topology, and (c) H: dim((n) ! H: dim((), if H: dim(() 1. On the other hand, we give examples showing the dimensio...
متن کاملOn the Maskit Slice of 4-dimensional Kleinian Punctured Torus Groups
Let Γ be a 3-dimensional Kleinian punctured torus group with accidental parabolics. The deformation space of Γ in the group of Möbius transformations on the 2-sphere is well-known as the Maskit slice of punctured torus groups. In this paper, we study the deformation space of Γ in the group of Möbius transformations on the 3-sphere, where Γ is naturally regarded as a 4-dimensional Kleinian group...
متن کامل