Simple groups without lattices

Counting lattices in simple Lie groups: the positive characteristic case

In this article we prove a conjecture of A. Lubotzky: let G = G0(K), where K is a local field of characteristic p ≥ 5, G0 is a simply connected, absolutely almost simple K-group of K-rank at least 2. We give the rate of growth of ρx(G) := |{Γ ⊆ G|Γ a lattice in G, vol(G/Γ) ≤ x}/ ∼ |, where Γ1 ∼ Γ2 if and only if there is an abstract automorphism θ of G such that Γ2 = θ(Γ1). We also study the ra...

Simple axiomatization of reticulations on residuated lattices

‎We give a simple and independent axiomatization of reticulations on residuated lattices‎, ‎which were axiomatized by five conditions in [C‎. ‎Mureşan‎, ‎The reticulation of a residuated lattice‎, ‎Bull‎. ‎Math‎. ‎Soc‎. ‎Sci‎. ‎Math‎. ‎Roumanie‎ ‎51 (2008)‎, ‎no‎. ‎1‎, ‎47--65]‎. ‎Moreover‎, ‎we show that reticulations can be considered as lattice homomorphisms between residuated lattices and b...

Distributive Lattices of λ-simple Semirings

In this paper, we study the decomposition of semirings with a semilattice additive reduct. For, we introduce the notion of principal left $k$-radicals $Lambda(a)={x in S | a stackrel{l}{longrightarrow^{infty}} x}$ induced by the transitive closure $stackrel{l}{longrightarrow^{infty}}$ of the relation $stackrel{l}{longrightarrow}$ which induce the equivalence relation $lambda$. Again non-transit...

Designs, Groups and Lattices

Free Groups in Lattices

Let G be any locally compact metrizable group. The main result of this paper, roughly stated, is that if F < G is any finitely generated free group and Γ < G any lattice, then up to a small perturbation and passing to a finite index subgroup, F is a subgroup of Γ. If G/Γ is noncompact then we require additional hypotheses that include G = SO(n, 1). MSC: 20E07, 20F65, 20F67, 22D40, 20E05.