Related papers: $n$-Kazhdan groups and higher spectral expanders

We prove some estimates for elementary symmetric polynomials on $\mathbb D^n.$ We show that these estimates are sharp which allow us to study the properties of closed symmetrized polydisc $\Gamma_n.$ Furthermore, we show the existence and…

We investigate the behavior of the higher-order degrees, db_n, of a finitely presented group G. These db_n are functions from H^1(G;Z) to Z whose values are the degrees certain higher-order Alexander polynomials. We show that if def(G) is…

This work provides an effective algorithm for distinguishing finite quotients between two non-isomorphic finitely generated Fuchsian groups $\Gamma$ and $\Lambda$. It will suffice to take a finite quotient which is abelian, dihedral, a…

Let $\Gamma<\mathrm{PSL}_2(\mathbb{C})\simeq \mathrm{Isom}^+(\mathbb{H}^3)$ be a finitely generated non-Fuchsian Kleinian group whose ordinary set $\Omega=\mathbb{S}^2-\Lambda$ has at least two components. Let $\rho : \Gamma \to…

In this paper, we study spectral gaps of closed hyperbolic surfaces for large genus. We show that for any fixed $k\geq 1$, as the genus goes to infinity, the maximum of $\lambda_k-\lambda_{k-1}$ over any thick part of the moduli space of…

In this article, we show that for a quasicompact scheme $X$ and $n>0,$ the $n$-th $K$-group $K_{n}(X)$ is a $\lambda$-module over a $\lambda$-ring $K_{0}(X)$ in the sense of Hesselholt.

In this article we improve the known Kazhdan constant for $SL_n(Z)$ with respect to the generating set of the elementary matrices. We prove that the Kazhdan constant is bounded from below by $[42\sqrt{n}+860]^{-1}$, which gives the exact…

Let $S=\{p_1, \dots, p_r,\infty\}$ for prime integers $p_1, \dots, p_r.$ Let $X$ be an $S$-adic compact nilmanifold, equipped with the unique translation invariant probability measure $\mu.$ We characterize the countable groups $\Gamma$ of…

We study the cokernel of the application given by the Cartan Matrix $C_\Lambda$ of a finite dimensional $k$-algebra $\Lambda.$ This produces a finitely generated abelian group, the Cartan group $G_\Lambda,$ which is invariant under derived…

In this article we extend results of Zomorrodian to determine upper bounds for the order of a nilpotent group of automorphisms of a complex $d$-dimensional family of compact Riemann surfaces, where $d \geqslant 1.$ We provide conditions…

For an $(n-1)$-Auslander algebra $\Lambda$ with global dimension $n$, we give some necessary conditions for $\Lambda$ admitting a maximal $(n-1)$-orthogonal subcategory in terms of the properties of simple $\Lambda$-modules with projective…

Let $\mathcal{F}_h^i(k,n)$ be the $i$th ordered configuration space of all distinct points $H_1,\ldots,H_h$ in the Grassmannian $Gr(k,n)$ of $k$-dimensional subspaces of $\mc^n$, whose sum is a subspace of dimension $i$. We prove that…

For any $n\geq 7$, $k\geq 3$, we give pairs of compact flat $n$-manifolds $M, M'$ with holonomy groups $\mathbb Z_2^k$, that are strongly isospectral, hence isospectral on $p$-forms for all values of $p$, having nonisomorphic cohomology…

For a field $F$ and a given integer $n>1$, Goncharov has given a complex $\Gamma_F(n)$ which he calls motivic and which he expects to rationally compute the weight $n$ motivic cohomology of $\text{Spec }F$, and hence its algebraic…

Let $F/{\mathbb Q}_p$ be a finite field extension, let $k$ be a field of characteristic $p$. Fix a Lubin Tate group $\Phi$ for $F$ and let $\Gamma\times\cdots\times\Gamma$ with $\Gamma={\mathcal O}_F^{\times}$ act on…

We present a method for constructing families of isospectral systems, using linear representations of finite groups. We focus on quantum graphs, for which we give a complete treatment. However, the method presented can be applied to other…

We classify all the $2$-arc-transitive strongly regular graphs, and use this classification to study the family of finite $(G,3)$-geodesic-transitive graphs of girth $4$ or $5$ for some group $G$ of automorphisms. For this application we…

In recent work, we showed that for all $q\in\tfrac{1}{2}\mathbb{Z}\setminus\mathbb{Z}_{\leq0}$ the sequence $\left\{\Gamma^{\left(n\right)}\left(q\right)\right\} _{n\geq1}$ contains transcendental elements infinitely often, with the density…

Let $X$ be a compact Hausdorff space, let $\Gamma$ be a discrete group that acts continuously on $X$ from the right, define $\widetilde{X} = \{(x,\gamma) \in X \times \Gamma : x\cdot\gamma= x\}$, and let $\Gamma$ act on $\widetilde{X}$ via…

Suppose $\Gamma$ is an arithmetic group defined over a global field $K$, that the $K$-type of $\Gamma$ is $A_n$ with $n \geq 2$, and that the ambient semisimple group that contains $\Gamma$ as a lattice has at least two noncocompact…