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 Higher Rank Wavelets

2011 ◽  
Vol 63 (3) ◽  
pp. 689-720
Author(s):  
Sean Olphert ◽  
Stephen C. Power
Keyword(s):  
Orthonormal Basis ◽  
Fourier Transforms ◽  
Latin Square ◽  
Haar Wavelets ◽  
Scaling Functions ◽  
Wavelet Sets ◽  
Compactly Supported ◽  
Rank 2 ◽  
Higher Rank ◽  
Meyer Wavelet

Abstract A theory of higher rank multiresolution analysis is given in the setting of abelian multiscalings. This theory enables the construction, from a higher rank MRA, of finite wavelet sets whose multidilations have translates forming an orthonormal basis in L2(ℝd). While tensor products of uniscaled MRAs provide simple examples we construct many nonseparable higher rank wavelets. In particular we construct Latin square wavelets as rank 2 variants of Haar wavelets. Also we construct nonseparable scaling functions for rank 2 variants of Meyer wavelet scaling functions, and we construct the associated nonseparable wavelets with compactly supported Fourier transforms. On the other hand we show that compactly supported scaling functions for biscaled MRAs are necessarily separable.

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.

scholarly journals AUTOMORPHIC ORBITS IN FREE GROUPS: WORDS VERSUS SUBGROUPS

2010 ◽  
Vol 20 (04) ◽  
pp. 561-590 ◽  
Author(s):  
PEDRO V. SILVA ◽  
PASCAL WEIL
Keyword(s):  
Free Group ◽  
Free Groups ◽  
Finitely Generated ◽  
Primitive Elements ◽  
Group Of Automorphisms ◽  
Rational Subset ◽  
Rank 2 ◽  
Higher Rank

We show that the following problems are decidable in a rank 2 free group F2: Does a given finitely generated subgroup H contain primitive elements? And does H meet the orbit of a given word u under the action of G, the group of automorphisms of F2? Moreover, decidability subsists if we allow H to be a rational subset of F2, or alternatively if we restrict G to be a rational subset of the set of invertible substitutions (a.k.a. positive automorphisms). In higher rank, the following weaker problem is decidable: given a finitely generated subgroup H, a word u and an integer k, does H contain the image of u by some k-almost bounded automorphism? An automorphism is k-almost bounded if at most one of the letters has an image of length greater than k.

Locally split and locally finite twin buildings of 2-spherical type

1999 ◽  
Vol 1999 (511) ◽  
pp. 119-143 ◽  
Author(s):  
Bernhard Mühlherr
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Algebraic Groups ◽  
Spherical Type ◽  
Locally Finite ◽  
Arbitrary Rank ◽  
Rank 2 ◽  
Twin Buildings

Abstract We prove a fixed point theorem for twin buildings of arbitrary rank. This theorem is then used to construct certain twin buildings whose existence was conjectured in [12]. As a consequence we obtain a classification of twin buildings whose rank 2 residues correspond to split algebraic groups over a field of cardinality at least 4. A similar result follows for twin buildings whose rank 2 residues are finite.

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 Ample vector bundles on a rational surface (higher rank)

1977 ◽  
Vol 66 ◽  
pp. 77-88
Author(s):  
Toshio Hosoh
Keyword(s):  
Vector Bundle ◽  
Vector Bundles ◽  
Rational Surface ◽  
Sufficient Condition ◽  
Chern Classes ◽  
Stable Vector Bundle ◽  
Rank 2 ◽  
Higher Rank

In the previous paper [1], we showed that the set of simple vector bundles of rank 2 on a rational surface with fixed Chern classes is bounded and we gave a sufficient condition for an H-stable vector bundle of rank 2 on a rational surface to be ample. In this paper, we shall extend the results of [1] to the case of higher rank.

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 Strong Banach property (T) for simple algebraic groups of higher rank

2014 ◽  
Vol 06 (01) ◽  
pp. 75-105 ◽  
Author(s):  
Benben Liao
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Algebraic Groups ◽  
Isometric Action ◽  
Cocompact Lattice ◽  
Uniform Embedding ◽  
The Family ◽  
Property T ◽  
Higher Rank ◽  
Finite Quotients

We extend Vincent Lafforgue's results to Sp4. As applications, the family of expanders constructed by finite quotients of a lattice in such a group does not admit a uniform embedding in any Banach space of type > 1, and any affine isometric action of such a group, or of any cocompact lattice in it, in a Banach space of type > 1 has a fixed point.

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