in this paper, we study the notion of solvable $l$-subgroup of an $l$-group and provide its level subset characterization and this justifies the suitability of this extension. throughout this work, we have used normality of an $l$-subgroup of an $l$-group in the sense of wu rather than liu.

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...

A Shimura surface is the quotient of either the product H×H of two hyperbolic planes or the unit ball HC in C by an irreducible arithmetic lattice. Examples include the normal quasiprojective varieties associated with the Hilbert and Picard modular groups, along with the solutions to many moduli problems for principally polarized abelian varieties. Special amongst the immersed projective algebr...

Let L be a Lie group and A a lattice in L. Suppose G is a non-compact simple Lie group realized as a Lie subgroup of L and G---A = L. Let a~G be such that Ada is semisimple and not contained in a compact subgroup of Aut (Lie(G)). Consider the expanding horospherical subgroup of G associated to a defined as U + = {g~ G:a-"ga"~ e as n ~ oo}. Let f2 be a non-empty open subset of U § and nf-* ~ be ...

Let Γ be an arithmetic lattice in a semisimple algebraic group over a number field. We show that if Γ has the congruence subgroup property, then the number of n-dimensional irreducible representations of Γ grows like n, where α is a rational number.

Self-assembly is ubiquitous in nature, particularly in biology, where it underlies the formation of protein quaternary structure and protein aggregation. Quaternary structure assembles deterministically and performs a wide range of important functions in the cell, whereas protein aggregation is the hallmark of a number of diseases and represents a nondeterministic self-assembly process. Here we...

We study infinite translation surfaces which are Z-covers of compact translation surfaces. We obtain conditions ensuring that such surfaces have Veech groups which are Fuchsian of the first kind and give a necessary and sufficient condition for recurrence of their straight-line flows. Extending results of Hubert and Schmithüsen, we provide examples of infinite non-arithmetic lattice surfaces, a...

We introduce a combinatorial version of Stallings-Bestvina-Feighn-Dunwoody folding sequences. We then show how they are useful in analyzing the solvability of the uniform subgroup membership problem for fundamental groups of graphs of groups. Applications include coherent right-angled Artin groups and coherent solvable groups.

A well-known result of universal algebra states: every algebraic lattice is isomorphic to the congruence lattice of an algebra. It is natural to ask whether every finite lattice occurs as the congruence lattice of a finite algebra. This fundamental question, asked over 45 years ago, is among the most elusive problems of universal algebra. Call a finite lattice L “representable” if it is the con...

A bstract We provide evidence for partial deconfinement — the of a SU( M ) subgroup N gauge group by using lattice Monte Carlo simulations. take matrix models as concrete examples. By appropriately fixing gauge, we observe that × submatrices deconfine. This gives direct at strong coupling. discuss applications to QCD and holography.