We introduce the co-surface graph CS of a finitely generated free group F and use it to study the geometry of hyperbolic group extensions of F. Among other things, we show that the Gromov boundary of the co-surface graph is equivariantly homeomorphic to the space of free arational F–trees and use this to prove that a finitely generated subgroup of Out(F) quasi-isometrically embeds into the co-s...

A celebrated theorem of Marshall Hall implies that finitely generated free groups are subgroup separable and that all of their finitely generated subgroups are retracts of finite-index subgroups. We use topological techniques inspired by the work of Stallings to prove that all limit groups share these two properties. This answers a question of Sela and provides a purely topological proof that f...

A celebrated theorem of Marshall Hall implies that finitely generated free groups are subgroup separable and that all of their finitely generated subgroups are retracts of finite-index subgroups. We use topological techniques inspired by the work of Stallings to prove that all limit groups share these two properties. This answers a question of Sela and provides a purely topological proof that f...

For a homomorphism f : A → B of commutative rings, letD(A,B) denote the kernel of the map Pic(A) → Pic(B). Let k be a field and assume that A is a finitely generated k-algebra. We prove a number of finiteness results for D(A,B). Here are four of them. [1]: Suppose B is a finitely generated and faithfully flat A-algebra which is geometrically integral over k. If k is perfect, we find that D(A,B)...

A canonical form for elements of a lattice freely generated by a partial lattice is given. This form agrees with Whitman’s canonical form for free lattices when the partial lattice is an antichain. The connection between this canonical form and the arithmetic of the lattice is given. For example, it is shown that every element of a finitely presented lattice has only finitely many minimal join ...

With the standard inference rule set for Nuprl, the type Atom cannot be proved either to be finite or infinite, despite the fact that any character string (over a certain finite alphabet) can be used to form a canonical expression for a member of the type. For each k ∈ N one can prove in the logic a formula to the effect that there are at least k atoms. One can do this simply by exhibiting k di...

This is a paper in mathematics, specifically in set theory. On the example of the measure recognition problem (MRP), the paper highlights the phenomenon of the utility of a multidisciplinary mathematical approach to a single mathematical problem, in particular, the value of a set-theoretic analysis. MRP asks if for a given Boolean algebra, B, and a property, Phi, of measures, one can recognize ...

The statement of the Hanna Neumann Conjecture (HNC) is purely algebraic: for a free group Γ and any nontrivial finitely generated subgroups A and B of Γ, rk (A ∩B)− 1 ≤ (rkA− 1)(rkB − 1). The goal of this paper is to systematically develop machinery that would allow for generalizations of HNC and to exhibit their relations with topology and analysis. On the topological side we define immersions...

We are interested in constructing concrete independent events in purely atomic probability spaces with geometric distribution. Among other facts we prove that there are uncountable many sequences of independent events.

decomposability of an algebraic structure into the :union: of its sub-structures goes back to g. scorza's theorem of 1926 for groups. an analogue of this theorem for rings has been recently studied by a. lucchini in 2012. on the study of this problem for non-group semigroups, the first attempt is due to clifford's work of 1961 for the regular semigroups. since then, n.p. mukherjee in ...