A note on weak conjugacy for homeomorphisms of surfaces
We explore the relation of weak conjugacy in the group of homeomorphisms isotopic to the identity, for surfaces.
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arXiv subjects
Maxime Wolff
Publications and source records attributed to Maxime Wolff.
Automorphisms of some variants of fine graphs
Recently Bowden, Hensel and Webb defined the fine curve graph for surfaces, extending the notion of curve graphs for the study of homeomorphism or diffeomorphism groups of surfaces. Later Long, Margalit, Pham, Verberne and Yao proved that for a closed surface of genus $g\geqslant 2$, the automorphism group of the fine graph is naturally isomorphic to the homeomorphism group of the surface. We extend this result to the torus case $g=1$; in fact our method works for more general surfaces, compact or not, orientable or not. We also discuss the case of a smooth version of the fine graph.
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Transitivity of normal subgroups of the mapping class group on character varieties
We prove that the action of any non-trivial normal subgroup of the mapping class group of a surface of genus $g\geqslant 2$ is almost minimal on the character variety $X(\pi_1\Sigma_g,{\rm SU}_2)$: the orbit of almost every point is dense.
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A basis for the Kauffman skein module of the product of a surface and a circle
The Kauffman bracket skein module $S(M)$ of a 3-manifold $M$ is a $\mathbb{Q}(A)$-vector space spanned by links in $M$ modulo the so-called Kauffman relations. In this article, for any closed oriented surface $\Sigma$ we provide an explicit spanning family for the skein modules $S(\Sigma\times S^1)$. Combined with earlier work of Gilmer and Masbaum, we answer their question about the dimension of $S(\Sigma\times S^1)$ being $2^{2g+1} + 2g -1$.
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Reconstructing maps out of groups
We show that, in many situations, a homeomorphism $f$ of a manifold $M$ may be recovered from the (marked) isomorphism class of a finitely generated group of homeomorphisms containing $f$. As an application, we relate the notions of {\em critical regularity} and of {\em differentiable rigidity}, give examples of groups of diffeomorphisms of 1-manifolds with strong differential rigidity, and in so doing give an independent, short proof of a recent result of Kim and Koberda that there exist finitely generated groups of $C^\alpha$ diffeomorphisms of a 1-manifold $M$, not embeddable into $\mathrm{Diff}^\beta(M)$ for any $\beta > \alpha > 1$.
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Rigidity of mapping class group actions on $S^1$
The mapping class group $\mathrm{Mod}_{g, 1}$ of a surface with one marked point can be identified with an index two subgroup of $\mathrm{Aut}(\pi_1 \Sigma_g)$. For a surface of genus $g \geq 2$, we show that any action of $\mathrm{Mod}_{g, 1}$ on the circle is either semi-conjugate to its natural action on the Gromov boundary of $\pi_1 \Sigma_g$, or factors through a finite cyclic group. For $g \geq 3$, all finite actions are trivial. This answers a question of Farb.
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The parallelogram identity on groups and deformations of the trivial character in SL_2(C)
We describe on any finitely generated group G the space of maps G->C which satisfy the parallelogram identity, f(xy)+f(xy^{-1})=2f(x)+2f(y). It is known (but not well-known) that these functions correspond to Zariski-tangent vectors at the trivial character of the character variety of G in SL_2(C). We study the obstructions for deforming the trivial character in the direction given by f. Along the way, we show that the trivial character is a smooth point of the character variety if dim H_1(G,C)<2 and not a smooth point if dim H_1(G,C)>2.
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A characterization of Fuchsian actions by topological rigidity
We prove that any rigid representation of $\pi_1\Sigma_g$ in $\mathrm{Homeo}_+(S^1)$ with Euler number at least $g$ is necessarily semi-conjugate to a discrete, faithful representation into $\mathrm{PSL}(2,\mathbb{R})$. Combined with earlier work of Matsumoto, this precisely characterizes Fuchsian actions by a topological rigidity property. Though independent, this work can be read as an introduction to the companion paper {\em rigidity and geometricity for surface group actions on the circle}, by the same authors.
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Rigidity and geometricity for surface group actions on the circle
We prove that rigid representations of the fundamental group of a surface into the group of oreintation-preserving homeomorphisms of the circle are geometric, thereby establishing a converse statement of a theorem by the first author.
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Dynamical rigidity of non discrete representations in PSL(2,R)
The aim of this note is to advertise on a result, not stated explicitly, but proved, in arXiv:0802.0512. Namely, if $\Gamma$ is any group, if $\rho_1$, $\rho_2$ are representations of $\Gamma$ in $\mathrm{PSL}(2,\mathbb{R})$, one of them being non elementary and non discrete, and if for all $\gamma\in\Gamma$, $\rho_1(\gamma)$ and $\rho_2(\gamma)$ have the same rotation number, then $\rho_1$ and $\rho_2$ are conjugate in $\mathrm{PSL}(2,\mathbb{R})$. In particular, if two non discrete, non elementary representations yield semi-conjugate actions on the circle, then they are conjugate in $\mathrm{PSL}(2,\mathbb{R})$.
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Six-point configurations in the hyperbolic plane and ergodicity of the mapping class group
Let $X$ be the space of isometry classes of ordered sextuples of points in the hyperbolic plane such that the product of the six corresponding rotations of angle $π$ is the identity. This space $X$ is closely related to the PSL$_2(\mathbb{R})$-character variety of the genus 2 surface $Σ$. In this article we study the topology and the natural symplectic structure on $X$, and we describe the action of the mapping class group of $Σ$ on $X$. This completes the classification of the ergodic components of the character variety in genus 2 initiated in our previous work.
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Compact anti-de Sitter 3-manifolds and folded hyperbolic structures on surfaces
We prove that any nonabelian, non-Fuchsian representation of a surface group into PSL(2,R) is the holonomy of a folded hyperbolic structure on the surface. Using similar ideas, we establish that any non-Fuchsian representation rho of a surface group into PSL(2,R) is strictly dominated by some Fuchsian representation j, in the sense that the hyperbolic translation lengths for j are uniformly larger than for rho; conversely, any Fuchsian representation j strictly dominates some non-Fuchsian representation rho, whose Euler class can be prescribed. This has applications to compact anti-de Sitter 3-manifolds.
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The modular action on PSL(2,R)-characters in genus 2
We explore the dynamics of the action of the mapping class group in genus 2 on the PSL(2,R)-character variety. We prove that this action is ergodic on the connected components of Euler class 1 and -1, as it was conjectured by Goldman. In the connected component of Euler class 0 there are two invariant open subsets, on one of them the action is ergodic. In this process we give a partial answer to a question of Bowditch.
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Connected components of the compactification of representation spaces of surface groups
The Thurston compactification of Teichmuller spaces has been generalized to many different representation spaces by J. Morgan, P. Shalen, M. Bestvina, F. Paulin, A. Parreau and others. In the simplest case of representations of fundamental groups of closed hyperbolic surfaces in PSL(2,R), we prove that this compactification is very degenerated: the nice behaviour of the Thurston compactification of the Teichmuller space contrasts with wild phenomena happening on the boundary of the other connected components of these representation spaces. We prove that it is more natural to consider a refinement of this compactification, which remembers the orientation of the hyperbolic plane. The ideal points of this compactification are fat R-trees, i.e., R-trees equipped with a planar structure.
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Non-injective representations of a closed surface group into $PSL(2,\mathbb R)$
Let $e$ denote the Euler class on the space $Hom(Γ_g, PSL(2,\mathbb R))$ of representations of the fundamental group $Γ_g$ of the closed surface $Σ_g$ of genus $g$. Goldman showed that the connected components of $Hom(Γ_g, PSL(2,\mathbb R))$ are precisely the inverse images $e^{-1}(k)$, for $2-2g\leq k\leq 2g-2$, and that the components of Euler class $2-2g$ and $2g-2$ consist of the injective representations whose image is a discrete subgroup of $PSL(2,\mathbb R)$. We prove that non-faithful representations are dense in all the other components. We show that the image of a discrete representation essentially determines its Euler class. Moreover, we show that for every genus and possible corresponding Euler class, there exist discrete representations.
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Framed holonomic knots
A holonomic knot is a knot in 3-space which arises as the 2-jet extension of a smooth function on the circle. A holonomic knot associated to a generic function is naturally framed by the blackboard framing of the knot diagram associated to the 1-jet extension of the function. There are two classical invariants of framed knot diagrams: the Whitney index (rotation number) W and the self linking number S. For a framed holonomic knot we show that W is bounded above by the negative of the braid index of the knot, and that the sum of W and |S| is bounded by the negative of the Euler characteristic of any Seifert surface of the knot. The invariant S restricted to framed holonomic knots with W=m, is proved to split into n, where n is the largest natural number with 2n < |m|+1, integer invariants. Using this, the framed holonomic isotopy classification of framed holonomic knots is shown to be more refined than the regular isotopy classification of their diagrams.
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