Zoom link： https://us06web.zoom.us/j/2715345558?pwd=eXRTTExpOVg4ODFYellsNXZVVlZvQT09
Schedule
.Date: 30 Apr., 2024
Speaker: Antonio Trusiani (Chalmers University of Technology）
Title: Singular cscK metrics on smoothable varieties
Abstract: We extend the notion of cscK metrics to singular varieties. We establish the existence of these canonical metrics on Q-Gorenstein smoothable klt varieties when the Mabuchi functional is coercive, these arise as a limit of cscK metrics on close-by fibres. The proof relies on developing a novel strong topology of pluripotential theory in families and establishing uniform estimates for cscK metrics. A key point is the lower semicontinuity of the coercivity threshold of Mabuchi functional along degenerate families of normal compact Kähler varieties with klt singularities. The latter suggests the openness of (uniform) K-stability for general polarized families of normal projective varieties. This is a joint work with Chung-Ming Pan and Tat Dat Tô.
.Date: 7 May, 2024
Speaker: Jesús A. Álvarez López (University of Santiago de Compostela (USC))
Title: A trace formula for foliated flows
Abstract: In the lecture, I will try to explain the ideas of a recent paper on the trace formula for foliated flows, written in collaboration with Yuri Kordyukov and Eric Leichtnam. Let $\mathcal{F}$ be a transversely oriented foliation of codimension one on a closed manifold $M$, and let $\phi=\{\phi^t\}$ be a foliated flow on $(M,\mathcal{F})$ (it maps leaves to leaves). Assume the closed orbits of $\phi$ are simple and its preserved leaves are transversely simple. In this case, there are finitely many preserved leaves, which are compact. Let $M^0$ denote their union, and write $M^1=M\setminus M^0$ and $\mathcal{F}^1=\mathcal{F}|_{M^1}$. We consider two locally convex Hausdorff spaces, $I(\mathcal{F})$ and $I'(\mathcal{F})$, consisting of the leafwise currents on $M$ that are conormal and dual-conormal to $M^0$, respectively. They become topological complexes with the differential operator $d_{\mathcal{F}}$ induced by the de~Rham derivative on the leaves, and they have an $\mathbb{R}$-action $\phi^*=\{\phi^{t\,*}\}$ induced by $\phi$. Let $\bar H^\bullet I(\mathcal{F})$ and $\bar H^\bullet I'(\mathcal{F})$ denote the corresponding leafwise reduced cohomologies, with the induced $\mathbb{R}$-action $\phi^*=\{\phi^{t\,*}\}$. The spaces $\bar H^\bullet I(\mathcal{F})$ and $\bar H^\bullet I'(\mathcal{F})$ are shown to be the central terms of short exact sequences in the category of continuous linear maps between locally convex spaces, where the other terms are described using Witten's perturbations of the de~Rham complex on $M^0$ and leafwise Witten's perturbations for $\mathcal{F}^1$. This is used to define some kind of Lefschetz distribution $L_{\rm dis}(\phi)$ of the actions $\phi^*$ on both $\bar H^\bullet I(\mathcal{F})$ and $\bar H^\bullet I'(\mathcal{F})$, whose value is a distribution on $\mathbb{R}$. Its definition involves several renormalization procedures, the main one is the b-trace of some smoothing b-pseudodifferential operator on the compact manifold with boundary obtained by cutting $M$ along $M^0$. We also prove a trace formula describing $L_{\rm dis}(\phi)$ in terms of infinitesimal data from the closed orbits and preserved leaves. This solves a conjecture of C.~Deninger involving two leafwise reduced cohomologies instead of a single one.
Remark (by Deng): This talk is based on arXiv:2402.06671
.Date: 14 May, 2024 (No Talks)
Past Talks
.Date: 23 Apr., 2024
Speaker: Giovanni Gentili (Università di Torino)
Title: Special metrics in hypercomplex Geometry
Abstract: The existence and search of special metrics plays a remarkably important role in Differential Geometry. After a brief overview of basic definitions and facts in hypercomplex Geometry, we will discuss certain notions of special metrics in the hypercomplex setting, focusing on the consequences of their existence. We will also introduce an Einstein condition for hyperhermitian metrics and describe the similarities with the Kähler-Einstein case. The talk is based on a joint work with Elia Fusi.
Date: 16 April 2024
Speaker: Simion Filip (Chicago)
Title: The volume of a divisor and cusp excursions of geodesics in hyperbolic manifolds
Abstract: The volume of a divisor on an algebraic variety measures the growth rate of the dimension of the space of sections of tensor powers of the associated line bundle. In the case of certain Calabi-Yau N-folds possessing a large group of pseudo-automorphisms, we show that the behavior of the volume can be highly oscillatory as the divisor class approaches the boundary of the pseudo-effective cone. This is explained by relating the volume function to the dynamical behavior of geodesics on certain hyperbolic manifolds.
After providing an introduction and some context for the above notions, I will discuss some of the ideas that go into the proof. (joint work with John Lesieutre and Valentino Tosatti)
.Date: 9 Apr., 2024
Speaker: Charles Cifarelli (Stony Brook)
Title: Non-compact Kähler-Ricci solitons, toric fibrations, and weighted K-stability
Abstract: In recent years, it has been shown that various problems in Kähler geometry can be unified under the framework of the weighted cscK equation, introduced by Lahdili. One feature of this setup is that, if a given manifold Y admits a special kind of fibration structure with toric fiber M and base B, then many interesting equations for the metric on Y can be reduced to the weighted cscK equation on M. This can be thought of as a generalized form of the Calabi Ansatz, which one recovers by taking M = \mathbb{C}. I will present on recent work extending some of this picture to the non-compact setting, with particular attention on (shrinking) Kähler-Ricci solitons. In particular, if a non-compact toric manifold M admits a weighted cscK metric (for suitable choices of weights), then it must be weighted K-stable. I will explain the relationship with Kähler-Ricci solitons, and time permitting I will explain how this leads to a simple proof of a well-known result of Futaki-Wang on the existence of shrinking Kähler-Ricci solitons on the total space of certain line bundles over a compact Kähler-Einstein Fano base.
.Date: 2 Apr., 2024
Speaker: Tomoyuki Hisamoto (Tokyo Metropolitan University)
Title: Quantization of the Kähler-Ricci flow, the entropy, and the optimal degeneration for a Fano manifold
Abstract: In recent years the study of K-unstable Fano manifolds atracts people's attention. In this talk I will introduce the geometric quantization of the Kähler-Ricci flow and the associated entropy functional introduced by Weiyong He. The "quantized entropy" coincides with the terminology in the quantum information theory. We also show some convergence results and discuss about the finite-dimensional analogue of the optimal degeneration.
.Date: 26 Mar., 2024
Speaker: Jian Song (Rutgers University)
Title: Minimal slopes for complex Hessian type equations
Abstract: The existence of smooth solutions to a broad class of complex Hessian type equations is related to nonlinear Nakai type criteria on intersection numbers on Kahler manifolds. Such a Nakai criteria can be interpreted as a slope stability condition analogous to the slope stability for Hermitian vector bundles over Kahler manifolds. In the present work, we initiate a program to find canonical solutions to such equations in the unstable case when the Nakai criteria fails. Conjecturally such solutions should arise as limits of natural parabolic flows and should be minimisers of the corresponding moment-map energy functionals. We implement our approach for the J-equation and the deformed Hermitian Yang-Mills equation on surfaces and some examples with symmetry. We further present the bubbling phenomena for the J-equation by constructing minimizing sequences of the moment-map energy functionals.
.Date: 19 Mar., 2024
Speaker: Yaxiong Liu (University of Maryland)
Title: The eigenvalue problem of complex Hessian operators
Abstract: In a very recent pair of nice papers of Badiane and Zeriahi, they consider the eigenvalue problem of complex Monge-Ampere and complex Hessian, and show that the C^{1,\bar{1}}-regularity of eigenfunction for MA and C^alpha-regularity for complex Hessian. They posed a question about the C^{1,1}-regularity. We give a positive answer and show the C^{1,1}-regularity and uniqueness of the eigenfunction. We also derive a number of applications, including a bifurcation-type theorem and geometric bounds for the eigenvalue. This is a joint work with Jianchun Chu and Nicholas McCleerey.
.Date: 12 Mar., 2024 (Additional talks)
Speaker: Thorsten Hertl (Albert-Ludwigs-Universität Freiburg) 19:30-20:00 (Beijing Time)
Title: Moduli Spaces of Positive Curvature Metrics
Abstract: Besides the space of positive scalar curvature metrics, various moduli spaces have gained a lot of attention. Among those, the observer moduli space arguably has the best behaviour from a homotopy-theoretical perspective because the subgroup of observer diffeomorphisms acts freely on the space of Riemannian metrics if the underlying manifold is connected.
In this talk, I will present how to construct non-trivial elements in the second homotopy of the observer moduli space of positive scalar curvature metrics for a large class for four-manifolds. I will further outline how to adapt this construction to produce the first non-trivial elements in higher homotopy groups of the observer moduli space of positive sectional curvature metrics on complex projective spaces.
Speaker: Zhicheng Han (Georg-August-Universität) 20:00-20:30 (Beijing Time)
Title: Spectra of Lie groups and application to L^2-invariants
Abstract: In this talk, I will explore the Laplace operator and Dirac operator on semisimple Lie groups. While the parallel problem on symmetric spaces has been well-studied in the last century, the corresponding problem is much less understood in general homogeneous spaces. We will examine the obstacles in extending existing techniques and discuss how some of them can be resolved in the case of group manifolds. Towards the end, we will see how the spectra data shall aid in computing certain topological L^2-invariants.
.Date: 12 Mar., 2024 21:00-22:00 (Beijing Time)
Speaker: Vyacheslav Lysov (London Institute for Mathematical Sciences)
Title: Chern-Gauss-Bonnet theorem via BV localization
Abstract: I will give a brief introduction to supersymmetric localization, BV formalism, and BV localization. I will show that the Euler class integral is a partition function for a zero-dimensional field theory with on-shell supersymmetry. I will describe the partition function as a BV integral, and deform the Lagrangian sub-manifold to evaluate the same integral as a sum over critical points for the Morse function.
Speaker: Ruadhaí Dervan (University of Glasgow)
Title: The universal structure of moment maps in complex geometry
Abstract: Much of complex geometry is motivated by linking the existence of solutions to geometric PDEs (producing "canonical metrics") to stability conditions in algebraic geometry. I will discuss a more basic question: what is the recipe to actually produce interesting geometric PDEs in complex geometry? The construction will be geometric, using a combination of universal families and tools from equivariant differential geometry. This is joint work with Michael Hallam.
Speaker: Xin Wang (KIAS)
Title: 5D Wilson Loops and Topological Strings on Fano-threefolds
Abstract: Geometric engineering provides a rich class of 5D supersymmetric gauge theories with eight supercharges, arising from M-theory compactification on non-compact Calabi-Yau threefolds. The counting of BPS states in the low-energy gauge theory is determined by the degeneracies of M2-branes wrapping holomorphic two-cycles in the Calabi-Yau threefold X. These degeneracies can also be calculated from the (refined) topological strings on the same manifold X. In this talk, I will explore the BPS spectrum of the 5D gauge theory with the insertion of a half-BPS Wilson loop operator. Utilizing M-theory realization, we derive the BPS expansion of the expectation value of the Wilson loop operator in terms of BPS sectors. It is found that the BPS sectors can be realized and computed from topological string theories. In the unrefined limit, the BPS sectors act as generating functions for Gromov-Witten invariants on compact (semi)-Fano threefolds constructed from X. We further conjecture a new refined holomorphic anomaly equations for the generating functions of the Wilson loops. These equations can completely solve the refined BPS invariants of Wilson loops for local \mathbb{P}^2 and local \mathbb{P}^1\times\mathbb{P}^1.
Speaker: Abdellah Lahdili (University of Aarhus) (Last talk of this semester)
Title: The Einstein-Hilbert functional and K-stability.
Abstract: In this talk, I will discuss a correspondence between the problem of finding a constant scalar curvature K\"ahler metric (cscK for short) on a polarized complex manifold and the CR-Yamabe problem on the associated circle bundle. I will then use a CR version of the Einstein-Hilbert functional to show K-semistability of polarised complex manifolds admitting cscK metrics. This is a joint work with Eveline Legendre and Carlo Scarpa.
Speaker: Denis Auroux (Harvard)
Title: Holomorphic discs and extended deformations in mirror symmetry
Abstract: Given a Lagrangian torus fibration on the complement of an anticanonical divisor in a Kahler manifold, one usually constructs a mirror space by gluing local charts (moduli spaces of local systems on generic torus fibers) via wall-crossing transformations determined by counts of Maslov index 0 holomorphic discs; this mirror also comes equipped with a regular function (the superpotential) which enumerates Maslov index 2 holomorphic discs. However, holomorphic discs of negative Maslov index deform this picture by introducing inconsistencies in the wall-crossing transformations; the geometric features of the resulting mirror can be understood in the language of extended deformations of Landau-Ginzburg models. We illustrate this phenomenon (and show that it actually occurs) on an explicit example (a 4-fold obtained by blowing up a Calabi-Yau toric variety), and discuss a family Floer approach to the geometry of the corrected mirror in this setting.
Speaker: Yuta Kusakabe (Kyoto University )
Title: Oka tubes in holomorphic line bundles
Abstract: A complex manifold is called an Oka manifold if continuous maps from Stein manifolds can be deformed into holomorphic maps with approximation and interpolation. Oka manifolds are characterized by a certain kind of ellipticity which is the opposite of Kobayashi-Eisenman-Brody hyperbolicity. Based on the fact that the zero section of a negative line bundle on a compact complex manifold admits a basis of pseudo-Kobayashi hyperbolic neighborhoods, we discuss the question of when the zero section of a line bundle admits a basis of Oka neighborhoods. This talk is based on joint work with Franc Forstnerič.
Speaker: Timothée Bénard (Warwick Mathematic Institute)
Title: Limit theorems on nilpotent Lie groups.
Abstract: I will present limit theorems for random walks on nilpotent Lie groups, obtained in a recent work with Emmanuel Breuillard. Most works on the topic assumed the step distribution of the walk to be centered in the abelianization of the group. Our main contribution is to authorize non-centered step distributions. In this case, new phenomena emerge: the large-scale geometry of the walk depends on the increment average, and the limit measure in the central limit theorem may not have full support in the ambient group.
Speaker: Martin de Borbon (Loughborough University)
Title: Bubbling of Kahler-Einstein metrics: examples and conjectures
Abstract: I will report on joint work with Cristiano Spotti. Chi Li's theory of normalized volumes permits the algebraic determination of tangent cones of singular Kahler-Einstein metrics. The general principle is to extend this to the dynamic setting of families and determine all possible rescaled limits using algebraic geometry. I will discuss examples in low dimensions, including polyhedral metrics on the sphere and Kahler-Einstein metrics on complex surfaces with cone singularities along curves.
Speaker: Marc Troyanov (Institut de Mathématiques EPFL)
Title: Double Forms, Curvature Integrals and the Gauss-Bonnet Formula
Abstract: In this talk I will briefly present the historical development behind Chern's proof in 1944 of the higher dimensional Gauss-Bonnet formula, starting with the work of H. Hopf back in 1925. I will then introduce the notion of double forms (which basically are sums of tensor products of differential forms) and explain how these tensors help us reformulate the Gauss-Bonnet-Chern-Formula and in particular clarify the meaning of Chern's boundary term. I will give some application and if time permits, some perspectives on non compact and singular Riemannian manifolds.
Speaker: Fernando Galaz-García (Durham University)
Title: Decompositions of 3-dimensional Alexandrov spaces
Abstract: Alexandrov spaces (with curvature bounded below) are metric generalizations of complete Riemannian manifolds with a uniform lower sectional curvature bound. Instances of Alexandrov spaces include compact Riemannian orbifolds and orbit spaces of isometric compact Lie group actions on compact Riemannian manifolds. In addition to being objects of intrinsic interest, Alexandrov spaces play an important role in Riemannian geometry, for example, in Perelman's proof of the Poincaré Conjecture. In this talk, I will discuss the topology and geometry of 3-dimensional Alexandrov spaces, focusing on extensions of basic results in 3-manifold topology (such as the prime decomposition theorem) to general three-dimensional Alexandrov spaces. This is joint work with Luis Atzin Franco Reyna, José Carlos Gómez-Larrañaga, Luis Guijarro, and Wolfgang Heil.
Speaker: Jeffrey D. Streets (University of California, Irvine) (22:00-23:00 Beijing Time !)
Title: Formal structure of scalar curvature in generalized Kahler geometry
Abstract: The Fujiki-Donaldson moment map formulation of scalar curvature problem, and the attendant Mabuchi-Semmes-Donaldson geometry of a Kahler class, play a central role in addressing the existence and uniqueness of constant scalar curvature Kahler metrics. Generalized Kahler (GK) geometry is a natural extension of Kahler geometry arising from Hitchin’s generalized geometry program and mathematical physics, and forms a particularly well-structured extension of Kahler geometry. Recently Goto defined a notion of scalar curvature in GK geometry as the moment map of a particular Hamiltonian action on the space of generalized Kahler structures. In this talk I will describe joint work with Vestislav Apostolov and Yury Ustinovskiy where we give an explicit description of the scalar curvature, and define a natural generalization of the Mabuchi-Semmes-Donaldson metric, leading to a Calabi-Lichnerowicz-Matsushima obstruction, generalizations of Futaki’s invariants, and a conditional uniqueness result.
Speaker: Gonçalo Oliveira (Instituto Superior Técnicon Lisbon)
Title: Einstein metrics from Derdziński duality
Abstract: (joint work with Rosa Sena-Dias) A theorem of Derdziński from the 1980's establishes that certain Einstein metrics are conformal to Bach-flat extremal Kahler metrics. Using this result Rosa Sena-Dias and I classified conformally Kähler, U(2)-invariant, Einstein metrics on the total space of O(−m). This yields infinitely many 1-parameter families of metrics exhibiting several different behaviors including asymptotically hyperbolic metrics (more specifically of Poincaré type), ALF metrics, and metrics which compactify to a Hirzebruch surface with a cone singularity along the ''divisor at infinity''. As an application of these results, we find many interesting phenomena. For instance, we exhibit the Taub-bolt Ricci-flat ALF metric as a limit of cone angle Einstein metrics on the blow up of CP2 at a point (in the limit when the cone angle converges to zero). We also construct Einstein metrics which are asymptotically hyperbolic and conformal to a scalar-flat Kähler metric and cannot be obtained by applying Derdziński's theorem.
Speaker: Gabriel Katz (MIT)
Title: Recovering contact forms from boundary data
Abstract: Let $X$ be a compact connected smooth manifold with boundary. The paper deals with contact $1$-forms $\beta$ on $X$, whose Reeb vector fields $v_\b$ admit Lyapunov functions $f$. We prove that any odd-dimensional $X$ admits such a contact form.
We tackle the question: how to recover $X$ and $\beta$ from the appropriate data along the boundary $\partial X$? We describe such boundary data and prove that they allow for a reconstruction of the pair $(X, \beta)$, up to a diffeomorphism of $X$. We use the term ``holography" for the reconstruction. We say that objects or structures inside $X$ are {\it holographic}, if they can be reconstructed from their $v_\b$-flow induced ``shadows" on the boundary $\partial X$.
For a given $\beta$, we study the contact vector fields $u$ on $X$ that preserve it invariant. Integrating $u$, we get a $1$-parameter family of contactomorphisms $\{\Phi^t(u)\}_{t \in \mathbb R}$ which maps Reeb trajectories to Reeb trajectories. This leads to estimates from below of the number of $\Phi^t(u)$-fixed Reeb trajectories.
We also introduce numerical invariants that measure how ``wrinkled" the boundary $\partial X$ is with respect to the $v_\beta$-flow and study their holographic properties under the contact forms preserving embeddings of equidimensional contact manifolds with boundary. We get some ``non-squeezing results" about such contact embedding, which are reminiscent of Gromov's non-squeezing theorem in symplectic geometry.
Speaker: Jacopo Stoppa (Scuola Internazionale Superiore di Studi Avanzati)
Title: K-stability and large complex structure limits
Abstract: According to mirror symmetry, the geometry of a given Fano manifold endowed with some extra data, including an arbitrary Kähler class, should be reflected in a mirror Landau-Ginzburg model, i.e. a noncompact complex manifold endowed with a nonconstant holomorphic function. On the other hand, a fundamental notion for constructing moduli of Fano manifolds is K-polystability, i.e. positivity of the Donaldson-Futaki invariants for nonproduct test-configurations. In this talk I will introduce the problem of characterising K-polystable Kähler classes on a Fano in terms of their mirror Landau-Ginzburg models. I will then discuss some first concrete results in the case of slope stability for del Pezzo surfaces. The computations involve the particular “large complex structure limit” of the Landau-Ginzburg model corresponding to scaling the Kähler class on the Fano, which acts trivially on K-polystability.
Speaker: Mario Garcia Fernandez (Universidad Autónoma de Madrid)
Title: Futaki invariants and harmonic metrics for the Hull-Strominger system
Abstract: The Hull-Strominger system has been proposed as a geometrization tool for understanding the moduli space of Calabi-Yau threefolds with topology change (‘Reid’s Fantasy’). In this talk we will introduce obstructions to the existence of solutions for these equations which combine infinite-dimensional moment maps, ‘harmonic metrics’, and a holomorphic version of generalized geometry. We will discuss the implications of these new obstructions in relation to a conjecture by S.-T. Yau for the Hull-Strominger system. Based on joint work with Raúl Gonzalez Molina, in arXiv:2303.05274 and arXiv:2301.08236.
Speaker: James Bonifacio (University of Mississippi)
Title: Spectral bounds on hyperbolic 3-manifolds
Abstract: I will discuss some new bounds on the spectra of Laplacian operators on hyperbolic 3-manifolds. One example of such a bound is that the spectral gap of the Laplace-Beltrami operator on a closed orientable hyperbolic 3-manifold must be less than 47.32, or less than 31.57 if the first Betti number is positive. The bounds are derived using two approaches, both of which employ linear programming techniques: 1) the Selberg trace formula, and 2) identities derived from the associativity of spectral decompositions. The second approach is inspired by an area of physics called the conformal bootstrap. This talk is based on work done in collaboration with Dalimil Mazáč and Sridip Pal.
Speaker: Javier Martínez-Aguinaga (Universidad Complutense de Madrid)
Title: $h$-Principle for maximal growth distributions
Abstract: The existence and classification problem for maximal growth distributions on smooth manifolds has garnered much interest in the mathematical community in recent years. Prototypical examples of maximal growth distributions are contact structures on $3$-dimensional manifolds and Engel distributions on $4$-dimensional manifolds.
The existence and classification of maximal growth distributions on open manifolds follows from Gromov’s $h$-Principle for open manifolds. Nonetheless, not so much was known for the case of closed manifolds and several open questions have been posed in the literature throughout the years about this problem.
In our recent work [1] we show that maximal growth distributions of rank $> 2$ abide by a full h-principle in all dimensions, providing thus a classification result from a homotopic viewpoint. As a consequence we answer in the positive, for $k > 2$, the long-standing open question posed by M. Kazarian and B. Shapiro more than 25 years ago of whether any parallelizable manifold admits a $k$-rank distribution of maximal growth.
[1]. Martínez-Aguinaga, Javier. Existence and classification of maximal growth distributions . Preprint. arXiv:2308.10762
Speaker: Timothée Bénard (Warwick Mathematic Institute)
Title: Limit theorems on nilpotent Lie groups.
Abstract: I will present limit theorems for random walks on nilpotent Lie groups, obtained in a recent work with Emmanuel Breuillard. Most works on the topic assumed the step distribution of the walk to be centered in the abelianization of the group. Our main contribution is to authorize non-centered step distributions. In this case, new phenomena emerge: the large-scale geometry of the walk depends on the increment average, and the limit measure in the central limit theorem may not have full support in the ambient group.
Speaker: Kajal Das (Indian Statistical Institute)
Title: On super-rigidity of Gromov's random monster group （Slides)
Abstract: In my talk, I will speak on the super-rigidity of Gromov's random monster group. It is a finitely generated random group $\Gamma_\alpha$ ( $\alpha$ is in a probability space $\mathcal{A}$) constructed using an expander graph by M. Gromov in 2000. It provides a counterexample to the Baum-Connes conjecture for groups with coefficients in commutative $C^*$-algebra. It is already known that it has global fixed point property for isometric affine action on $L^p$ spaces for $1< p<\infty$ (in particular, Property (T) ) for a.e. $\alpha$ due to Gromov and Naor-Silberman. It is also hyperbolically rigid, i.e., any isometric action of the group on a Gromov-hyperbolic space is elementary for a.e. $\alpha$ (due to Gruber-Sisto-Tessera). In this talk, I will discuss the following type of super-rigidity problem (motivated by Margulis super-rigidity theorem): for which countable group $G$, any collection of homomorphisms $\phi_\alpha:\Gamma_\alpha\rightarrow G$ have a finite image for a.e. $\alpha$? This question was first addressed for linear groups by Naor-Silberman. The super-rigidity follows immediately from the literature for groups with a-$L^p$-menablity and K-amenable groups. In this talk, we will show that $\Gamma_\alpha$ has super-rigidity with respect to the following groups $G$: mapping class group $MCG(S_{g,p})$, braid group $B_n$, automorphism group of a free group $Aut(F_n)$, outer automorphism group of a free group $Out(F_N)$ . We will also show a stability result of the class of groups $G$.
