No Arabic abstract
We study the Chabauty compactification of two families of closed subgroups of $SL(n,mathbb{Q}_p)$. The first family is the set of all parahoric subgroups of $SL(n,mathbb{Q}_p)$. Although the Chabauty compactification of parahoric subgroups is well studied, we give a different and more geometric proof using various Levi decompositions of $SL(n,mathbb{Q}_p)$. Let $C$ be the subgroup of diagonal matrices in $SL(n, mathbb{Q}_p)$. The second family is the set of all $SL(n,mathbb{Q}_p)$-conjugates of $C$. We give a classification of the Chabauty limits of conjugates of $C$ using the action of $SL(n,mathbb{Q}_p)$ on its associated Bruhat--Tits building and compute all of the limits for $nleq 4$ (up to conjugacy). In contrast, for $ngeq 7$ we prove there are infinitely many $SL(n,mathbb{Q}_p)$-nonconjugate Chabauty limits of conjugates of $C$. Along the way we construct an explicit homeomorphism between the Chabauty compactification in $mathfrak{sl}(n, mathbb{Q}_p)$ of $SL(n,mathbb{Q}_p)$-conjugates of the $p$-adic Lie algebra of $C$ and the Chabauty compactification of $SL(n,mathbb{Q}_p)$-conjugates of $C$.
In this paper we describe the local limits under conjugation of all closed connected subgroups of $SL_3(mathbb{R})$ in the Chabauty topology.
We give a quadratic lower bound on the dimension of the space of conjugacy classes of subgroups of SL(n,R) that are limits under conjugacy of the diagonal subgroup. We give the first explicit examples of abelian n-1 dimensional subgroups of SL(n,R) which are not such a limit, however all such abelian groups are limits of the diagonal group iff n < 5.
Fugledes conjecture in $mathbb{Q}_p$ is proved. That is to say, a Borel set of positive and finite Haar measure in $mathbb{Q}_p$ is a spectral set if and only if it tiles $mathbb{Q}_p$ by translation.
We generalize Bonahon and Wongs $mathrm{SL}_2(mathbb{C})$-quantum trace map to the setting of $mathrm{SL}_3(mathbb{C})$. More precisely, for each non-zero complex number $q$, we associate to every isotopy class of framed oriented links $K$ in a thickened punctured surface $mathfrak{S} times (0, 1)$ a Laurent polynomial $mathrm{Tr}_lambda^q(K) = mathrm{Tr}_lambda^q(K)(X_i^q)$ in $q$-deformations $X_i^q$ of the Fock-Goncharov coordinates $X_i$ for a higher Teichm{u}ller space, depending on the choice of an ideal triangulation $lambda$ of the surface $mathfrak{S}$. Along the way, we propose a definition for a $mathrm{SL}_n(mathbb{C})$-version of this invariant.
In this article, we prove that a compact open set in the field $mathbb{Q}_p$ of $p$-adic numbers is a spectral set if and only if it tiles $mathbb{Q}_p$ by translation, and also if and only if it is $p$-homogeneous which is easy to check. We also characterize spectral sets in $mathbb{Z}/p^n mathbb{Z}$ ($pge 2$ prime, $nge 1$ integer) by tiling property and also by homogeneity. Moreover, we construct a class of singular spectral measures in $mathbb{Q}_p$, some of which are self-similar measures.