Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userInvariant random subgroups over non-Archimedean local fields
116
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
Let $G$ be a higher rank semisimple linear algebraic group over a non-Archimedean local field. The simplicial complexes corresponding to any sequence of pairwise non-conjugate irreducible lattices in $G$ are Benjamini-Schramm convergent to the Bruhat-Tits building. Convergence of the relative Plancherel measures and normalized Betti numbers follows. This extends the work of Abert, Bergeron, Biringer, Gelander, Nokolov, Raimbault and Samet from real Lie groups to linear groups over arbitrary local fields. Along the way, various results concerning Invariant Random Subgroups and in particular a variant of the classical Borel density theorem are also extended.
rate research
Read More
In this paper, we study iterative methods on the coefficients of the rational univariate representation (RUR) of a given algebraic set, called global Newton iteration. We compare two natural approaches to define locally quadratically convergent iterations: the first one involves Newton iteration applied to the approximate roots individually and then interpolation to find the RUR of these approximate roots; the second one considers the coefficients in the exact RUR as zeroes of a high dimensional map defined by polynomial reduction, and applies Newton iteration on this map. We prove that over fields with a p-adic valuation these two approaches give the same iteration function, but over fields equipped with the usual Archimedean absolute value, they are not equivalent. In the latter case, we give explicitly the iteration function for both approaches. Finally, we analyze the parallel complexity of the differen
This paper concerns character sheaves of connected reductive algebraic groups defined over non-Archimedean local fields and their relation with characters of smooth representations. Although character sheaves were devised with characters of representations of finite groups of Lie type in mind, character sheaves are perfectly well defined for reductive algebraic groups over any algebraically closed field. Nevertheless, the relation between character sheaves of an algebraic group $G$ over an algebraic closure of a field $K$ and characters of representations of $G(K)$ is well understood only when $K$ is a finite field and when $K$ is the field of complex numbers. In this paper we consider the case when $K$ is a non-Archimedean local field and explain how to match certain character sheaves of a connected reductive algebraic group $G$ with virtual representations of $G(K)$. In the final section of the paper we produce examples of character sheaves of general linear groups and matching admissible virtual representations.
Let $X$ be a proper geodesic Gromov hyperbolic metric space and let $G$ be a cocompact group of isometries of $X$ admitting a uniform lattice. Let $d$ be the Hausdorff dimension of the Gromov boundary $partial X$. We define the critical exponent $delta(mu)$ of any discrete invariant random subgroup $mu$ of the locally compact group $G$ and show that $delta(mu) > frac{d}{2}$ in general and that $delta(mu) = d$ if $mu$ is of divergence type. Whenever $G$ is a rank-one simple Lie group with Kazhdans property $(T)$ it follows that an ergodic invariant random subgroup of divergence type is a lattice. One of our main tools is a maximal ergodic theorem for actions of hyperbolic groups due to Bowen and Nevo.
This paper provides an overview of recent progress on the interplay between tropical geometry and non-archimedean analytic geometry in the sense of Berkovich. After briefly discussing results by Baker, Payne and Rabinoff in the case of curves, we explain a result by Cueto, Habich and the author comparing the tropical Grassmannian of planes to the analytic Grassmannian. We also give an overview of the general higher-dimensional theory developed by Gubler, Rabinoff and the author. In particular, we explain the construction of generalized skeleta in which are polyhedral substructures of Berkovich spaces lending themselves to comparison with tropicalizations. We discuss the slope formula for the valuation of rational functions and explain two results on the comparison between polyhedral substructures of Berkovich spaces and tropicalizations.
For a positive integer $g$, let $mathrm{Sp}_{2g}(R)$ denote the group of $2g times 2g$ symplectic matrices over a ring $R$. Assume $g ge 2$. For a prime number $ell$, we give a self-contained proof that any closed subgroup of $mathrm{Sp}_{2g}(mathbb{Z}_ell)$ which surjects onto $mathrm{Sp}_{2g}(mathbb{Z}/ellmathbb{Z})$ must in fact equal all of $mathrm{Sp}_{2g}(mathbb{Z}_ell)$. The result and the method of proof are both motivated by group-theoretic considerations that arise in the study of Galois representations associated to abelian varieties.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
