Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userIntegrable tautness of isometries of complex hyperbolic spaces
128
0
0.0
(
0
)
Authors
Alessio Savini
Ask ChatGPT about the research
No Arabic abstract
Consider $n geq 2$. In this paper we prove that the group $text{PU}(n,1)$ is $1$-taut. This result concludes the study of $1$-tautness of rank-one Lie groups of non-compact type. Additionally the tautness property implies a classification of finitely generated groups which are $text{L}^1$-measure equivalent to lattices of $text{PU}(n,1)$. More precisely, we show that $text{L}^1$-measure equivalent groups must be extensions of lattices of $text{PU}(n,1)$ by a finite group.
rate research
Read More
We investigate the translation lengths of group elements that arise in random walks on weakly hyperbolic groups. In particular, without any moment condition, we prove that non-elementary random walks exhibit at least linear growth of translation lengths. As a corollary, almost every random walk on mapping class groups eventually becomes pseudo-Anosov and almost every random walk on $mathrm{Out}(F_n)$ eventually becomes fully irreducible. If the underlying measure further has finite first moment, then the growth rate of translation lengths is equal to the drift, the escape rate of the random walk. We then apply our technique to investigate the random walks induced by the action of mapping class groups on Teichmuller spaces. In particular, we prove the spectral theorem under finite first moment condition, generalizing a result of Dahmani and Horbez.
We prove that the Teichmuller space of surfaces with given boundary lengths equipped with the arc metric (resp. the Teichmuller metric) is almost isometric to the Teichmuller space of punctured surfaces equipped with the Thurston metric (resp. the Teichmuller metric).
Let $Gamma$ be a torsion-free lattice of $text{PU}(p,1)$ with $p geq 2$ and let $(X,mu_X)$ be an ergodic standard Borel probability $Gamma$-space. We prove that any maximal Zariski dense measurable cocycle $sigma: Gamma times X longrightarrow text{SU}(m,n)$ is cohomologous to a cocycle associated to a representation of $text{PU}(p,1)$ into $text{SU}(m,n)$, with $1 < m leq n$. The proof follows the line of Zimmer Superrigidity Theorem and requires the existence of a boundary map, that we prove in a much more general setting. As a consequence of our result, it cannot exist a maximal measurable cocycle with the above properties when $n eq m$.
We prove that the deformation space AH(M) of marked hyperbolic 3-manifolds homotopy equivalent to a fixed compact 3-manifold M with incompressible boundary is locally connected at minimally parabolic points. Moreover, spaces of Kleinian surface groups are locally connected at quasiconformally rigid points. Similar results are obtained for deformation spaces of acylindrical 3-manifolds and Bers slices.
We define the surface complex for $3$-manifolds and embark on a case study in the arena of Seifert fibered spaces. The base orbifold of a Seifert fibered space captures some of the topology of the Seifert fibered space, so, not surprisingly, the surface complex of a Seifert fibered space always contains a subcomplex isomorphic to the curve complex of the base orbifold.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
