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Register a new userLinear growth of translation lengths of random isometries on Gromov hyperbolic spaces and Teichmuller spaces
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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.
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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).
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.
We consider pseudo-Anosov mapping classes on a closed orientable surface of genus $g$ that fix a rank $k$ subgroup of the first homology of the surface. We first show that there exists a uniform constant $C>0$ so that the minimal asymptotic translation length on the curve complex among such pseudo-Anosovs is bounded below by $C over g(2g-k+1)$. This interpolates between results of Gadre-Tsai and of the first author and Shin, who treated the cases of the entire mapping class group ($k = 0$) and the Torelli subgroup ($k = 2g$), respectively. We also discuss possible strategy to obtain an upper bound. Finally, we construct a pseudo-Anosov on a genus $g$ surface whose maximal invariant subspace is of rank $2g-1$ and the asymptotic translation length is of $asymp 1/g$ for all $g$. Such pseudo-Anosovs are further shown to be unable to normally generate the whole mapping class groups. As Lanier-Margalit proved that pseudo-Anosovs with small translation lengths on the Teichmuller spaces normally generate mapping class groups, our observation provides a restriction on how small the asymptotic translation lengths on curve complexes should be if the similar phenomenon holds for curve complexes.
A family of coordinates $psi_h$ for the Teichmuller space of a compact surface with boundary was introduced in cite{l2}. In the work cite{m1}, Mondello showed that the coordinate $psi_0$ can be used to produce a natural cell decomposition of the Teichmuller space invariant under the action of the mapping class group. In this paper, we show that the similar result also works for all other coordinate $psi_h$ for any $h geq 0$.
We prove that the length spectrum metric and the arc-length spectrum metric are almost-isometric on the $epsilon_0$-relative part of Teichmuller spaces of surfaces with boundary.
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