Related papers: Super approximation for $\text{SL}_2\times \text{S…

Without using the $p$-adic Langlands correspondence, we prove that for many finite length smooth representations of $\mathrm{GL}_2(\mathbf{Q}_p)$ on $p$-torsion modules the $\mathrm{GL}_2(\mathbf{Q}_p)$-linear morphisms coincide with the…

We construct an explicit generating sets $F_n$ and $\tilde F_n$ of the alternating and the symmetric groups, which make the Cayley graphs $C(Alt(n), F_n)$ and $C(Sym(n), \tilde F_n)$ a family of bounded degree expanders for all sufficiently…

For the p-adic group G=SL (2) , we present results of the computations of the sums of the Bernstein projectors of a given depth. Motivation for the computations is based on a conversation with Roger Howe in August 2013. The computations are…

We show that strong approximate lattices in higher-rank semi-simple algebraic groups are arithmetic.

We begin an investigation of supersymmetric theories based on exceptional groups. The flat directions are most easily parameterized using their correspondence with gauge invariant polynomials. Symmetries and holomorphy tightly constrain the…

The super-algebraic structure of a generalized version of the Jaynes-Cummings model is investigated. We find that a Z2 graded extension of the so(2,1) Lie algebra is the underlying symmetry of this model. It is isomorphic to the…

We introduce the notions of symmetric and symmetrizable representations of $\text{SL}_2(\mathbb{Z})$. The linear representations of $\text{SL}_2(\mathbb{Z})$ arising from modular tensor categories are symmetric and have congruence kernel.…

We show that the normalized supercharacters of principal admissible modules, associated to each integrable atypical module over the affine Lie superalgebra $\widehat{sl}_{2|1}$ can be modified, using Zwegers' real analytic corrections, to…

The so--called subgroup commutativity degree $sd(G)$ of a finite group $G$ is the number of permuting subgroups $(H,K) \in \mathrm{L}(G) \times \mathrm{L}(G)$, where $\mathrm{L}(G)$ is the subgroup lattice of $G$, divided by…

We attack a conjecture of J. Rogawski: any cocompact lattice in $S U (2, 1)$ for which the ball quotient $X = B^2 / \Gamma$ satisfies $b_1 (X) = 0$ and $H^{1, 1} (X) \cap H^2 (X, \bbq) \approx \bbq$ is arithmetic. We prove the Archimedian…

It is well known that every closed Riemann surface $S$ of genus $g \geq 2$, admitting a group $G$ of conformal automorphisms so that $S/G$ has triangular signature, can be defined over a finite extension of ${\mathbb Q}$. It is interesting…

Let $\lambda$ be a primitive root of unity of order $\ell$. We introduce a family of finite-dimensional algebras $\{\mathcal{D}_{\lambda,N}(\mathfrak{sl}_2)\}_{N\in\mathbb{N}_0}$ over the complex numbers, such that…

Let G2 be the exceptional Lie group of automorphisms of the complex Cayley algebra and C be a smooth, connected, projective curve of genus at least 2. Using the map obtained from extension of structure groups, we prove explicit links…

We provide and motivate in this paper a natural framework for the study of approximate lattices. Namely, we consider approximate lattices in so-called $S$-adic linear groups and define relevant notions of arithmeticity. We also adapt to…

We construct a 2-generated group $\Gamma $ such that its Cayley graph possesses finite connected subsets with arbitrarily big finite Heesch number.

We start with definitions of the general notions of the theory of $\Bbb Z_{2}$-graded algebras. Then we consider theory of inductive families of $\Bbb Z_{2}$-graded semisimple finite-dimensional algebras and its representations in the…

We show that random Cayley graphs of finite simple (or semisimple) groups of Lie type of fixed rank are expanders. The proofs are based on the Bourgain-Gamburd method and on the main result of our companion paper, establishing strongly…

We consider a number of examples of groups together with an infinite conjugation invariant generating set, including: the free group with the generating set of all separable elements; surface groups with the generating set of all…

We construct motivic $\ell$-adic representations of $\GQ$ into exceptional groups of type $E_7,E_8$ and $G_2$ whose image is Zariski dense. This answers a question of Serre. The construction is uniform for these groups and uses the…

We propose a novel construction of finite hypergraphs and relational structures that is based on reduced products with Cayley graphs of groupoids. To this end we construct groupoids whose Cayley graphs have large girth not just in the usual…