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Register a new userThe behaviour of Fenchel-Nielsen distance under a change of pants decomposition
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Given a topological orientable surface of finite or infinite type equipped with a pair of pants decomposition $mathcal{P}$ and given a base complex structure $X$ on $S$, there is an associated deformation space of complex structures on $S$, which we call the Fenchel-Nielsen Teichmuller space associated to the pair $(mathcal{P},X)$. This space carries a metric, which we call the Fenchel-Nielsen metric, defined using Fenchel-Nielsen coordinates. We studied this metric in the papers cite{ALPSS}, cite{various} and cite{local}, and we compared it to the classical Teichmuller metric (defined using quasi-conformal mappings) and to another metric, namely, the length spectrum, defined using ratios of hyperbolic lengths of simple closed curves metric. In the present paper, we show that under a change of pair of pants decomposition, the identity map between the corresponding Fenchel-Nielsen metrics is not necessarily bi-Lipschitz. The results complement results obtained in the previous papers and they show that these previous results are optimal.
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Using tropical geometry, Mikhalkin has proved that every smooth complex hypersurface in $mathbb{CP}^{n+1}$ decomposes into pairs of pants: a pair of pants is a real compact $2n$-manifold with cornered boundary obtained by removing an open regular neighborhood of $n+2$ generic hyperplanes from $mathbb{CP}^n$. As is well-known, every compact surface of genus $ggeqslant 2$ decomposes into pairs of pants, and it is now natural to investigate this construction in dimension 4. Which smooth closed 4-manifolds decompose into pairs of pants? We address this problem here and construct many examples: we prove in particular that every finitely presented group is the fundamental group of a 4-manifold that decomposes into pairs of pants.
A Riemann surface $X$ is said to be of emph{parabolic type} if it supports a Greens function. Equivalently, the geodesic flow on the unit tangent of $X$ is ergodic. Given a Riemann surface $X$ of arbitrary topological type and a hyperbolic pants decomposition of $X$ we obtain sufficient conditions for parabolicity of $X$ in terms of the Fenchel-Nielsen parameters of the decomposition. In particular, we initiate the study of the effect of twist parameters on parabolicity. A key ingredient in our work is the notion of textit{non standard half-collar} about a hyperbolic geodesic. We show that the modulus of such a half-collar is much larger than the modulus of a standard half-collar as the hyperbolic length of the core geodesic tends to infinity. Moreover, the modulus of the annulus obtained by gluing two non standard half-collars depends on the twist parameter, unlike in the case of standard collars. Our results are sharp in many cases. For instance, for zero-twist flute surfaces as well as half-twist flute surfaces with concave sequences of lengths our results provide a complete characterization of parabolicity in terms of the length parameters. It follows that parabolicity is equivalent to completeness in these cases. Applications to other topological types such as surfaces with infinite genus and one end (a.k.a. the infinite Loch-Ness monster), the ladder surface, Abelian covers of compact surfaces are also studied.
Let $Sigma_g$ be a compact, connected, orientable surface of genus $g geq 2$. We ask for a parametrization of the discrete, faithful, totally loxodromic representations in the deformation space ${rm Hom}(pi_1(Sigma_g), {rm SU}(3,1))/{rm SU}(3,1)$. We show that such a representation, under some hypothesis, can be determined by $30g-30$ real parameters.
We construct a quasiconformally homogeneous hyperbolic Riemann surface-other than the hyperbolic plane-that does not admit a bounded pants decomposition. Also, given a connected orientable topological surface of infinite type with compact boundary components, we construct a complete hyperbolic metric on the surface that has bounded geometry but does not admit a bounded pants decomposition.
We investigate the terms arising in an identity for hyperbolic surfaces proved by Luo and Tan, namely showing that they vary monotonically in terms of lengths and that they verify certain convexity properties. Using these properties, we deduce two results. As a first application, we show how to deduce a theorem of Thurston which states, in particular for closed hyperbolic surfaces, that if a simple length spectrum dominates another, then in fact the two surfaces are isometric. As a second application, we show how to find upper bounds on the number of pairs of pants of bounded length that only depend on the boundary length and the topology of the surface.
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