We prove that the abstract commensurator of a nontrivial free group, an infinite surface group, or more generally a group that splits appropriately over a cyclic subgroup is not finitely generated. This applies in particular to all torsion-free word-hyperbolic groups with infinite outer automorphism group and abelianization of rank at least 2. We also construct a finitely generated group which ...

We construct first examples of infinite finitely generated residually finite torsion groups with positive rank gradient. In particular, these groups are non-amenable. Some applications to problems about cost and L-Betti numbers are discussed.

We prove the global well-posedness and we study the linear response for a system of two coupled equations composed of a Dirac equation for an infinite rank operator and a nonlinear wave or Klein-Gordon equation.

The space of marked commutative rings on n given generators is a compact metrizable space. We compute the Cantor-Bendixson rank of any member of this space. For instance, the Cantor-Bendixson rank of the free commutative ring on n generators is ωn, where ω is the smallest infinite ordinal. More generally, we work in the space of finitely generated modules over a given commutative ring.

We will use commutators to provide decompositions of 3×3 matrices as sums whose terms satisfy some polynomial identities, and we apply them bounded linear operators endomorphisms free modules infinite rank. In particular it is proved that every operator an infinite-dimensional complex Hilbert space a sum four automorphisms order 3 simple ring obtained quotient the endomorphism vector modulo its...

The goal of this paper is to give an explicit construction of the Fock spaces of the parafermion and the paraboson algebra, for an infinite set of generators. This is equivalent to constructing certain unitary irreducible lowest weight representations of the (infinite rank) Lie algebra so(∞) and of the Lie superalgebra osp(1|∞). A complete solution to the problem is presented, in which the Fock...

Young’s lattice of a partition λ consists of all partitions whose Ferrers diagrams fit inside λ. Several infinite families of partitions are given whose Young’s lattice is not rank unimodal. Some related problems are discussed.