The Hopf Property for Subgroups of Hyperbolic Groups

A group is said to be Hopfian if every surjective endomorphism of the group is injective. We show that finitely generated subgroups of torsion-free hyperbolic groups are Hopfian. Our proof generalizes a theorem of Sela (Topology 35 (2) 1999, 301–321).     

Congruences and Closure Endomorphisms of Hilbert Algebras

In this paper we describe the congruences of a finite Hilbert algebra in terms of its closure endomorphisms. We use this result to give a necessary and sufficient condition under which two finite Hilbert algebras share the same monoid of endomorphisms.     

On the classification and modular extendability of E0-semigroups on factors

In this paper we study modular extendability and equimodularity of endomorphisms and E₀-semigroups on factors, when these definitions are recast to the context of faithful normal semifinite weights and this dependency is analyzed. We show that modular extendability is a property that does not depend on the choice of weights, it is a cocycle conjugacy invariant and it is preserved under tensoring. Furthermore, we prove a necessary and sufficient condition for equimodularity of endomorphisms in the context of weights. This extends previously known results regarding the necessity of this condition in the case of states. The classification of E₀-semigroups on factors is considered: a modularly extendable E₀-semigroup is said to be of type EI, EII or EIII if its modular extension is of type I, II or III, respectively. We prove that all types exist on properly infinite factors. We show that q-CCR flows are not extendable, and we extend previous results by the first author regarding the non-extendability of CAR flow to a larger class of quasi-free states. We also compute the coupling index and the relative commutant index for the CAR flows and q-CCR flows. As an application, by considering repeated tensors of the CAR flows we show that there are infinitely many non cocycle conjugate non-extendable E₀-semigroups on the hyperfinite factors of types II₁ and III_λ, for $\text{◂+▸}\lambda \in (0,1)$.     

Accessibility of partially hyperbolic endomorphisms with 1D center-bundles

We prove that partially hyperbolic endomorphisms with one dimensional center-bundles and non-trivial unstable bundles are stably accessible. And there is residual subset $\Res$ of partially hyperbolic volume preserving endomorphisms with one dimensional center-bundles such that every $f \in \Res$ is stably accessible. In the end, we prove the accessibility of Gan''s example.     

Characterizations of totally ordered sets by their various endomorphisms

We characterize totally ordered sets within the class of all ordered sets containing at least three-element chains using a simple relationship between their isotone transformations and the so called 2-, 3-, 4-endomorphisms which are introduced in the paper. Another characterization of totally ordered sets within the class of ordered sets of a locally finite height with at least four-element chains in terms of the regular semigroup theory is also given.     

A note on cocycle-conjugate endomorphisms of von Neumann algebras

We show that two cocycle-conjugate endomorphisms of an arbitrary von Neumann algebra that satisfy certain stability conditions are conjugate endomorphisms, when restricted to some specific von Neumann subalgebras. As a consequence of this result, we obtain a new criterion for conjugacy of Powers shift endomorphisms acting on factors of type