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Register a new userAlmost isometries between Teichmuller spaces
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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).
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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.
We study relations between maps between relatively hyperbolic groups/spaces and quasisymmetric embeddings between their boundaries. More specifically, we establish a correspondence between (not necessarily coarsely surjective) quasi-isometric embeddings between relatively hyperbolic groups/spaces that coarsely respect peripherals, and quasisymmetric embeddings between their boundaries satisfying suitable conditions. Further, we establish a similar correspondence regarding maps with at most polynomial distortion. We use this to characterise groups which are hyperbolic relative to some collection of virtually nilpotent subgroups as exactly those groups which admit an embedding into a truncated real hyperbolic space with at most polynomial distortion, generalising a result of Bonk and Schramm for hyperbolic groups.
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.
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 $h geq 0$.
Let G and F be finitely generated groups with infinitely many ends and let A and B be graph of groups decompositions of F and G such that all edge groups are finite and all vertex groups have at most one end. We show that G and F are quasi-isometric if and only if every one-ended vertex group of A is quasi-isometric to some one-ended vertex group of B and every one-ended vertex group of B is quasi-isometric to some one-ended vertex group of A. From our proof it also follows that if G is any finitely generated group, of order at least three, the groups: G*G, G*Z,G*G*G and G* Z/2Z are all quasi-isometric.
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