Rigidity of Compact Fuchsian Manifolds with Convex Boundary

A compact Fuchsian manifold with boundary is a hyperbolic 3-manifold homeomorphic to $S_g \times [0; 1]$ such that the boundary component $S_g \times \{ 0\}$ is geodesic. We prove that a compact Fuchsian manifold with convex boundary is uniquely determined by the induced path metric on $S_g \times \{1\}$. We do not put further restrictions on the boundary except convexity.

ABELIAN CYCLES IN THE HOMOLOGY OF THE TORELLI GROUP

In the early 1980s, Johnson defined a homomorphism $\mathcal {I}_{g}^1\to \bigwedge ^3 H_1\left (S_{g},\mathbb {Z}\right )$ , where $\mathcal {I}_{g}^1$ is the Torelli group of a closed, connected, and oriented surface of genus g with a boundary component and $S_g$ is the corresponding surface without a boundary component. This is known as the Johnson homomorphism. We study the map induced by the Johnson homomorphism on rational homology groups and apply it to abelian cycles determined by disjoint bounding-pair maps, in order to compute a large quotient of $H_n\left (\mathcal {I}_{g}^1,\mathbb {Q}\right )$ in the stable range. This also implies an analogous result for the stable rational homology of the Torelli group $\mathcal {I}_{g,1}$ of a surface with a marked point instead of a boundary component. Further, we investigate how much of the image of this map is generated by images of such cycles and use this to prove that in the pointed case, they generate a proper subrepresentation of $H_n\left (\mathcal {I}_{g,1}\right )$ for $n\ge 2$ and g large enough.

The mapping class group action on -character varieties

Let $\unicode[STIX]{x1D6F4}$ be a compact orientable surface of genus $g=1$ with $n=1$ boundary component. The mapping class group $\unicode[STIX]{x1D6E4}$ of $\unicode[STIX]{x1D6F4}$ acts on the $\mathsf{SU}(3)$ -character variety of $\unicode[STIX]{x1D6F4}$ . We show that the action is ergodic with respect to the natural symplectic measure on the character variety.

Maximal green sequences for arbitrary triangulations of marked surfaces (Extended Abstract)

International audience In general, the existence of a maximal green sequence is not mutation invariant. In this paper we show that it is in fact mutation invariant for cluster quivers associated to most marked surfaces. We develop a procedure to find maximal green sequences for cluster quivers associated to an arbitrary triangulation of closed higher genus marked surfaces with at least two punctures. As a corollary, it follows that any triangulation of a marked surface with at least one boundary component has a maximal green sequence.

Microseismic Signal Denoising via Empirical Mode Decomposition, Compressed Sensing, and Soft-thresholding

Microseismic signal denoising is of great significance for P wave, S wave first arrival picking, source localization, and focal mechanism inversion. Therefore, an Empirical Mode Decomposition (EMD), Compressed Sensing (CS), and Soft-thresholding (ST) combined EMD_CS_ST denoising method is proposed. First, through EMD decomposition of the noise signal, a series of Intrinsic Mode Functions (IMF) from high frequency to low frequency are obtained. By calculating the correlation coefficient between each IMF and the original signal, the boundary component between the signal and the noise was identified, and the boundary component and its previous components were sparsely processed in the discrete wavelet transform domain to obtain the original sparse coefficient θ. Second, θ is filtered by ST to get the reconstruction coefficient θnew after denoising. Then, CS was used to recover and reconstruct θnew to get the denoised IMFnew component and then recombined with the remaining IMF components to get the signal after denoising. In the simulation experiment, the denoising process of EMD_CS_ST method is introduced in detail, and the denoising ability of EMD_CS_ST, DWT, EEMD, and VMD_DWT under 10 different noise levels is discussed. The signal-to-noise ratio, signal standard deviation, correlation coefficient, waveform diagram, and spectrogram before and after denoising are compared and analyzed. Moreover, the signals obtained from the underground cavern of the Shuangjiangkou hydropower station were denoised by the EMD_CS_ST method, and the signals before and after denoising were analyzed by time-frequency spectrum. These results show that the proposed method has better denoising ability.

On the quotients of mapping class groups of surfaces by the Johnson subgroups

We study quotients of mapping class groups ${\Gamma _{g,1}}$ of oriented surfaces with one boundary component by the subgroups ${{\cal I}_{g,1}}(k)$ in the Johnson filtrations, and we show that the stable classifying spaces ${\mathbb {Z}} \times B{({\Gamma _\infty }/{{\cal I}_\infty }(k))^ + }$ after plus-construction are infinite loop spaces, fitting into a tower of infinite loop space maps that interpolates between the infinite loop spaces ${\mathbb {Z}} \times B\Gamma _\infty ^ + $ and ${\mathbb {Z}} \times B{({\Gamma _\infty }/{{\cal I}_\infty }(1))^ + } \simeq {\mathbb {Z}} \times B{\rm{Sp}}{({\mathbb {Z}})^ + }$ . We also show that for each level k of the Johnson filtration, the homology of these quotients with suitable systems of twisted coefficients stabilises as the genus of the surface goes to infinity.

Johnson–Levine homomorphisms and the tree reduction of the LMO functor

Let $\mathcal{M}$ denote the mapping class group of Σ, a compact connected oriented surface with one boundary component. The action of $\mathcal{M}$ on the nilpotent quotients of π1(Σ) allows to define the so-called Johnson filtration and the Johnson homomorphisms. J. Levine introduced a new filtration of $\mathcal{M}$ , called the Lagrangian filtration. He also introduced a version of the Johnson homomorphisms for this new filtration. The first term of the Lagrangian filtration is the Lagrangian mapping class group, whose definition involves a handlebody bounded by Σ, and which contains the Torelli group. These constructions extend in a natural way to the monoid of homology cobordisms. Besides, D. Cheptea, K. Habiro and G. Massuyeau constructed a functorial extension of the LMO invariant, called the LMO functor, which takes values in a category of diagrams. In this paper we give a topological interpretation of the upper part of the tree reduction of the LMO functor in terms of the homomorphisms defined by J. Levine for the Lagrangian mapping class group. We also compare the Johnson filtration with the filtration introduced by J. Levine.

Short loops in surfaces with a circular boundary component

It is a classical theorem of Loewner that the systole of a Riemannian torus can be bounded in terms of its area. We answer a question of a similar flavor of Robert Young showing that if [Formula: see text] is a Riemannian surface with connected boundary in [Formula: see text], such that the boundary curve is a standard unit circle, then the length of the shortest non-contractible loop in [Formula: see text] is bounded in terms of the area of [Formula: see text].