Related papers: Maximal and Borel Anosov representations in $Sp(4,…

In this article, we examine the restriction of the metaplectic representation $\pi$ over a $p$-adic field $k$, $p\neq2$, of zero characteristic to an isotropic torus $S$ contained in the symplectic group. First we give necessary and…

Previously, Reynolds showed that any irreducible nonsurjective endomorphism can be represented by an irreducible immersion on a finite graph. We give a new proof of this and also show a partial converse holds when the immersion has…

We give an explicit description of the restriction of discrete series representations of SU(2,1) to a Borel subgroup and to a maximal exponential solvable subgroup and we interpret our results in the framework of orbit method, Hamiltonian…

A pseudo-Anosov homeomorphism of a surface is a canonical representative of its mapping class. In this paper, we explain that a transitive pseudo-Anosov flow is similarly a canonical representative of its stable Hamiltonian class. It…

In this article we construct a type of deformations of representations $\pi_1(M)\rightarrow G$ where $G$ is an arbitrary lie group and $M$ is a large class of manifolds including CAT(0) manifolds. The deformations are defined based on…

Let W be a compact manifold and let \rho be a representation of its fundamental group into PSL(2,C). The volume of \rho is defined by taking any \rho-equivariant map from the universal cover of W to H^3 and then by integrating the pull-back…

We prove that the fundamental group of any integer homology 3-sphere different from the 3-sphere admits irreducible representations of its fundamental group in SL(2,C). For hyperbolic integer homology spheres this comes with the definition,…

We give a formula for the character of the representation of the symmetric group $S_n$ on each isotypic component of the cohomology of the set of regular elements of a maximal torus of $SL_n$, with respect to the action of the centre.

We show that finitely generated, purely pseudo-Anosov subgroups of the fundamental groups of surface bundles over tori are convex cocompact as subgroups of the mapping class group via the Birman exact sequence. This generalizes the fact…

A polygonal surface in the pseudo-hyperbolic space H^(2,n) is a complete maximal surface bounded by a lightlike polygon in the Einstein universe Ein^(1,n) with finitely many vertices. In this article, we give several characterizations of…

Given a holomorphic or anti-holomorphic involution on a complex variety, the Smith inequality says that the total $\mathbb{F}_2$-Betti number of the fixed locus is no greater than the total $\mathbb{F}_2$-Betti number of the ambient…

We prove the rigidity of Witten-Reshetikhin-Turaev $\mathrm{SU}(2)$ and $\mathrm{SO}(3)$ quantum representations of mapping class groups at all prime levels for closed surfaces of genus at least $7$. The proof relies on Ocneanu rigidity of…

Springer varieties are studied because their cohomology carries a natural action of the symmetric group $S_n$ and their top-dimensional cohomology is irreducible. In his work on tangle invariants, Khovanov constructed a family of Springer…

We show that for any irreducible representation of $Sp_{4n}(F_{q})$, the subspace of all its $Sp_{2n}(F_{q^{2}})$-invariants is at most one-dimensional. In terms of Lusztig symbols, we give a complete list of irreducible unipotent…

Let X be a hyperbolic surface and H the fundamental group of a hyperbolic 3-manifold that fibers over the circle with fiber X. Using the Birman exact sequence, H embeds in the mapping class group Mod(Y) of the surface Y obtained by removing…

We compute the Markov convexity invariant of the continuous infinite dimensional Heisenberg group $\mathbb{H}_\infty$ to show that it is Markov 4-convex and cannot be Markov $p$-convex for any $p < 4$. As Markov convexity is a biLipschitz…

We prove that a word hyperbolic group whose Gromov boundary properly contains a $2$-sphere cannot admit a projective Anosov representation into $\mathsf{Sp}_{2m}(\mathbb{C})$, $m\in \mathbb{N}$. We also prove that a word hyperbolic group…

For every genuine irreducible admissible smooth representation $\pi$ of the metaplectic group $\widetilde{\Sp}(2n)$ over a p-adic field, and every smooth oscillator representation $\omega_\psi$ of $\widetilde{\Sp}(2n)$, we prove that the…

By providing new finite criteria which certify that a finitely generated subgroup of $\mathrm{SL}(d,\mathbb{R})$ or $\mathrm{SL}(d,\mathbb{C})$ is projective Anosov, we obtain a practical algorithm to verify the Anosov condition. We…

Recall that the group $PSL(2,\mathbb R)$ is isomorphic to $PSp(2,\mathbb R),\ SO_0(1,2)$ and $PU(1,1).$ The goal of this paper is to examine the various ways in which Fuchsian representations of the fundamental group of a closed surface of…