Stationary characters on lattices of semisimple Lie groups

We show that stationary characters on irreducible lattices $\Gamma < G$ Γ < G of higher-rank connected semisimple Lie groups are conjugation invariant, that is, they are genuine characters. This result has several applications in representation theory, operator algebras, ergodic theory and topological dynamics. In particular, we show that for any such irreducible lattice $\Gamma < G$ Γ < G , the left regular representation $\lambda _{\Gamma }$ λ Γ is weakly contained in any weakly mixing representation $\pi $ π . We prove that for any such irreducible lattice $\Gamma < G$ Γ < G , any Uniformly Recurrent Subgroup (URS) of $\Gamma $ Γ is finite, answering a question of Glasner–Weiss. We also obtain a new proof of Peterson’s character rigidity result for irreducible lattices $\Gamma < G$ Γ < G . The main novelty of our paper is a structure theorem for stationary actions of lattices on von Neumann algebras.

Prime, irreducible, and completely irreducible lattice preradicals on modular complete lattices (I)

Based on the concept of a lattice preradical recently introduced in [T. Albu and M. Iosif, Lattice preradicals with applications to Grothendieck categories and torsion theories, J. Algebra 444 (2015) 339–366], we present and investigate in this paper, the latticial counterparts of the concepts of prime and irreducible preradicals on the category Mod-[Formula: see text] of all unital right [Formula: see text]-modules over an associative ring [Formula: see text] with identity, introduced and studied in [F. Raggi, J. Ríos Montes, H. Rincón, R. Fernández-Alonso and C. Signoret, Prime and irreducible preradicals, J. Algebra Appl. 4 (2005) 451–466].

Projective lattices of tiled orders

Tiled orders over discrete valuation ring have been studied since the 1970s by many mathematicians, in particular, by Yategaonkar V.A., Tarsy R.B., Roggenkamp K.W, Simson D., Drozd Y.A., Zavadsky A.G. and Kirichenko V.V. Yategaonkar V.A. proved that for every n > 2, there is, up to an isomorphism, a finite number of tiled orders over a discrete valuation ring O of finite global dimension which lie in $M_n(K)$ where K is a field of fractions of a commutatively discrete valuation ring O. The articles by R.B. Tarsy, V.A. Yategaonkar, H. Fujita, W. Rump and others are devoted to the study of the global dimension of tiled orders. H. Fujita described the reduced tiled orders in Mn(D) of finite global dimension for n = 4; 5. V.M. Zhuravlev and D.V. Zhuravlev described reduced tiled orders in Mn(D) of finite global dimension for n = 6: This paper examines the necessary condition for the finiteness of the global dimension of the tile order. Let A be a tiled order. The kernel of the projective resolvent of an irreducible lattice has the form M1f1 +M2f2 + ::: +Msfs, where Mi is irreducible lattice, fi is some vector. If the tile order has a finite global dimension, then there is a projective lattice that is the intersection of projective lattices. This condition is the one explored in the paper.

A Dehn function for Sp(2n, ℤ)

Gromov conjectured that any irreducible lattice in a symmetric space of rank at least [Formula: see text] should have at most polynomial Dehn function. We prove that the lattice [Formula: see text] has quadratic Dehn function when [Formula: see text]. By results of Broaddus, Farb, and Putman, this implies that the Torelli group in large genus is at most exponentially distorted.

Optimal natural dualities: the role of endomorphisms

The optimality of dualities on a quasivariety , generated by a finite algebra , has been introduced by Davey and Priestley in the 1990s. Since every optimal duality is determined by a transversal of a certain family of subsets of Ω, where Ω is a given set of relations yielding a duality on , an understanding of the structures of these subsets—known as globally minimal failsets—was required. A complete description of globally minimal failsets which do not contain partial endomorphisms has recently been given by the author and H. A. Priestley. Here we are concerned with globally minimal failsets containing endomorphisms. We aim to explain what seems to be a pattern in the way endomorphisms belong to these failsets. This paper also gives a complete description of globally minimal failsets whose minimal elements are automorphisms, when is a subdirectly irreducible lattice-structured algebra.

Congruence Normal Covers of Finitely Generated Lattice Varieties

We consider certain pseudovarieties K of lattices which are closed under the doubling of convex sets. For each such K, given an arbitrary finite lattice 𝓛, we describe the covers of the variety V(𝓛) of the form V(𝓛, K) with K a subdirectly irreducible lattice in K.

Equivariant images of projective space under the action of SL (n, ℤ)

The point of this note is to answer in the affirmative a question of G. A. Margulis. In the course of his proof of the finiteness of either the cardinality or the index of a normal subgroup of an irreducible lattice in a higher rank semi-simple Lie group [3], [4], Margulis proves that if Γ = SL (n, ℤ),n≥3, (X, μ) is a measurable Γ-space, μ quasi-invariant, and φ: ℙn−1→Xis a measure class preserving Γ-map, then either φ is a measure space isomorphism or μ is supported on a point. Margulis then asks whether the topological analogue of this result is true. This is answered in the following.