scholarly journals BOREL DENSITY FOR APPROXIMATE LATTICES

2019 ◽  
Vol 7 ◽  
Author(s):  
MICHAEL BJÖRKLUND ◽  
TOBIAS HARTNICK ◽  
THIERRY STULEMEIJER
Keyword(s):  
Dynamical Systems ◽  
Algebraic Groups ◽  
Local Fields ◽  
Density Theorems

We extend classical density theorems of Borel and Dani–Shalom on lattices in semisimple, respectively solvable algebraic groups over local fields to approximate lattices. Our proofs are based on the observation that Zariski closures of approximate subgroups are close to algebraic subgroups. Our main tools are stationary joinings between the hull dynamical systems of discrete approximate subgroups and their Zariski closures.

scholarly journals Random groups and nonarchimedean lattices

2014 ◽  
Vol 2 ◽  
Author(s):  
SYLVAIN BARRÉ ◽  
MIKAËL PICHOT
Keyword(s):  
Algebraic Groups ◽  
Local Fields ◽  
Density Model ◽  
Rank 2 ◽  
Higher Rank ◽  
Random Groups

AbstractWe consider models of random groups in which the typical group is of intermediate rank (in particular, it is not hyperbolic). These models are parallel to Gromov’s well-known constructions, and include for example a ‘density model’ for groups of intermediate rank. The main novelty is the higher rank nature of the random groups. They are randomizations of certain families of lattices in algebraic groups (of rank 2) over local fields.

scholarly journals CHABAUTY LIMITS OF ALGEBRAIC GROUPS ACTING ON TREES THE QUASI-SPLIT CASE

2018 ◽  
Vol 19 (4) ◽  
pp. 1031-1091
Author(s):  
Thierry Stulemeijer
Keyword(s):  
Algebraic Group ◽  
Closed Subset ◽  
Algebraic Groups ◽  
Local Fields ◽  
Locally Finite ◽  
Simple Algebraic Group ◽  
Groups Acting On Trees ◽  
Residue Characteristic

Given a locally finite leafless tree $T$, various algebraic groups over local fields might appear as closed subgroups of $\operatorname{Aut}(T)$. We show that the set of closed cocompact subgroups of $\operatorname{Aut}(T)$ that are isomorphic to a quasi-split simple algebraic group is a closed subset of the Chabauty space of $\operatorname{Aut}(T)$. This is done via a study of the integral Bruhat–Tits model of $\operatorname{SL}_{2}$ and $\operatorname{SU}_{3}^{L/K}$, that we carry on over arbitrary local fields, without any restriction on the (residue) characteristic. In particular, we show that in residue characteristic $2$, the Tits index of simple algebraic subgroups of $\operatorname{Aut}(T)$ is not always preserved under Chabauty limits.

RESIDUAL PROPERTIES OF FREE PRO-P GROUPS

2001 ◽  
Vol 33 (5) ◽  
pp. 578-582 ◽  
Author(s):  
YIFTACH BARNEA
Keyword(s):  
Positive Characteristic ◽  
Congruence Subgroup ◽  
Probabilistic Method ◽  
Algebraic Groups ◽  
Local Fields ◽  
Random Generation ◽  
Zero Characteristic ◽  
Residual Properties ◽  
Characteristic Case ◽  
Lie Methods

Recall that if S is a class of groups, then a group G is residually-S if, for any element 1 ≠ g ∈ G, there is a normal subgroup N of G such that g ∉ N and G/N ∈ S. Let Λ be a commutative Noetherian local pro-p ring, with a maximal ideal M. Recall that the first congruence subgroup of SLd(Λ) is: SL1d(Λ) = ker (SLd(Λ) → SLd(Λ/M)).Let K ⊆ ℕ. We define SΛ(K) = ∪d∈K{open subgroups of SL1d(Λ)}. We show that if K is infinite, then for Λ = [ ]p[[t]] and for Λ = ℤp a finitely generated non-abelian free pro-p group is residually-SΛ(K). We apply a probabilistic method, combined with Lie methods and a result on random generation in simple algebraic groups over local fields. It is surprising that the case of zero characteristic is deduced from the positive characteristic case.

On the topology of geometric and rational orbits for algebraic group actions over valued fields

2020 ◽  
Vol 23 (6) ◽  
pp. 965-981
Author(s):  
Phuong Bac Dao
Keyword(s):  
Algebraic Group ◽  
Local Field ◽  
Algebraic Groups ◽  
Local Fields ◽  
Unipotent Group ◽  
Affine Variety ◽  
Local Function ◽  
Relative Orbit ◽  
Completely Reducible ◽  
The Relationship

AbstractIn this note, we study the relationship between Zariski and relative closedness for actions of (smooth) algebraic groups defined over valued (mainly local) fields of any characteristic. In particular, we use some recent basic results regarding the completely reducible subgroups and cocharacter-closedness due to Bate–Herpel–Röhrle–Tange and Uchiyama to construct some actions of simple algebraic groups G of the types {D_{4}}, {E_{6}}, {E_{7}}, {E_{8}}, {G_{2}} on an affine variety defined over a local function field k, and {v\in V(k)} such that the geometric orbit {G.v} is Zariski closed although the corresponding relative orbit {G(k).v} is not closed in the topology induced from k. Besides, by using an interesting result due to Gabber, Gille and Moret-Bailly, we show that this phenomenon does not appear when we consider the action of either a smooth unipotent group or a smooth commutative algebraic group, defined over an admissible valued (e.g., local) field.

scholarly journals FROM THE FUNCTION-SHEAF DICTIONARY TO QUASICHARACTERS OF -ADIC TORI

2015 ◽  
Vol 17 (1) ◽  
pp. 1-37
Author(s):  
Clifton Cunningham ◽  
David Roe
Keyword(s):  
Cohomology Group ◽  
Monoidal Category ◽  
Algebraic Groups ◽  
Group Scheme ◽  
Local Fields ◽  
Locally Finite ◽  
Isomorphism Classes ◽  
Algebraic Tori ◽  
Component Group ◽  
Character Sheaves

We consider the rigid monoidal category of character sheaves on a smooth commutative group scheme $G$ over a finite field $k$, and expand the scope of the function-sheaf dictionary from connected commutative algebraic groups to this setting. We find the group of isomorphism classes of character sheaves on $G$, and show that it is an extension of the group of characters of $G(k)$ by a cohomology group determined by the component group scheme of $G$. We also classify all morphisms in the category character sheaves on $G$. As an application, we study character sheaves on Greenberg transforms of locally finite type Néron models of algebraic tori over local fields. This provides a geometrization of quasicharacters of $p$-adic tori.

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