Let G be a connected, solvable linear algebraic group over a number field K, let S be a finite set of places of K that contains all the infinite places, and let O(S) be the ring of S-integers of K. We define a certain closed subgroup GO(S) of GS = ∏ v∈S GKv that contains GO(S), and prove that GO(S) is a superrigid lattice in GO(S), by which we mean that finite-dimensional representations α : GO...

Let ω = (−1 + √ −3)/2. For any lattice P ⊆ Zn, P = P + ωP is a subgroup of On K , where OK = Z[ω] ⊆ C. As C is naturally isomorphic to R2, P can be regarded as a lattice in R2n. Let P be a multiplicative lattice (principal lattice or congruence lattice) introduced by Rosenbloom and Tsfasman. We concatenate a family of special codes with tP · (P + ωP ), where tP is the generator of a prime ideal...

Let G be a finite product of SL(2,Ki)′s for local fields Ki of characteristic zero. We present a discreteness criterion for non-solvable subgroups of G containing an irreducible lattice of a maximal unipotent subgroup of G. In particular such a subgroup has to be arithmetic. This extends a previous result of A. Selberg when G is a product of SL2(R)′s.

a subgroup $h$ is said to be $nc$-supplemented in a group $g$ if there exists a subgroup $kleq g$ such that $hklhd g$ and $hcap k$ is contained in $h_{g}$, the core of $h$ in $g$. we characterize the supersolubility of finite groups $g$ with that every maximal subgroup of the sylow subgroups is $nc$-supplemented in $g$.

suppose that $h$ is a subgroup of $g$‎, ‎then $h$ is said to be‎ ‎$s$-permutable in $g$‎, ‎if $h$ permutes with every sylow subgroup of‎ ‎$g$‎. ‎if $hp=ph$ hold for every sylow subgroup $p$ of $g$ with $(|p|‎, ‎|h|)=1$)‎, ‎then $h$ is called an $s$-semipermutable subgroup of $g$‎. ‎in this paper‎, ‎we say that $h$ is partially $s$-embedded in $g$ if‎ ‎$g$ has a normal subgroup $t$ such that $ht...

We present the first explicit connection between quantum computation and lattice problems. Namely, our main result is a solution to the Unique Shortest Vector Problem (SVP) under the assumption that there exists an algorithm that solves the hidden subgroup problem on the dihedral group by coset sampling. Additionally, we present an approach to solving the hidden subgroup problem on the dihedral...

Article outline This article gives a brief overview of recent developments in metric number theory, in particular, Diophantine approximation on manifolds, obtained by applying ideas and methods coming from dynamics on homogeneous spaces. Glossary 1. Definition: Metric Diophantine approximation 2. Basic facts 3. Introduction 4. Connection with dynamics on the space of lattices 5. Diophantine app...

We classify all locally finite joinings of a horospherical subgroup action on Γ\G when Γ is a Zariski dense geometrically finite subgroup of G = PSL2(R) or PSL2(C). This generalizes Ratner’s 1983 joining theorem for the case when Γ is a lattice in G. One of the main ingredients is equidistribution of non-closed horospherical orbits with respect to the Burger-Roblin measure which we prove in a g...

Let G be a locally compact group equipped with a left Haar measure μ, i.e. a Borel regular measure which is finite on compact, positive on open and invariant under left multiplications — by Haar’s theorem such μ exists and is unique up to normalization. The group G is called unimodular if μ is also right invariant, or equivalently if it is symmetric in the sense that μ(A) = μ(A−1) for every mea...

The article is dedicated to groups in which the set of abnormal and normal subgroups (U -subgroups) forms a lattice. A complete description of these groups under the additional restriction that every counternormal subgroup is abnormal is obtained.