The purpose of this note is to show that the collection LF of finite volume hyperbolic 3-manifolds whose fundamental groups contain a subgroup that is locally free but not free is commensurably infinite. This result is stated formally as Theorem 4.1, and gives a strong answer to a question of Kropholler given in the Problem List in Niblo and Roller [19]. Recall that two hyperbolic 3-manifolds N...

We had participants decide which one of two applicants was better qualified for a scholarship. They also judged the difference between them (comparative judgment). The applicants were described by features (grades) in different subjects (dimensions). The grades on some dimensions were missing (unique dimensions) for an alternative while all the grades were available on other dimensions (common ...

We propose a new framework that distinguishes among individual and situational factors in the social comparison process that produces competitive behavior. The familiar individual factors naturally vary among similarly situated people, including the relevance of the performance dimension, the commensurability of rivals, and their relationship closeness to the individual. Researchers have long e...

We introduce orbital functionals ∫ β simultaneously for each commensurability class of orbital surfaces. They are realized on infinitely dimensional orbital divisor spaces spanned by (arithmetic-geodesic real 2-dimensional) orbital curves on any orbital surface. We discover infinitely many of them on each commensurability class of orbital Picard surfaces, which are real 4-spaces with cusps and ...

This Essay offers a utilitarian perspective on Martha Nussbaum’s theory of justice. Nussbaum believes that society should guarantee to every individual a threshold level of central human capabilities. Although Nussbaum’s approach has considerable appeal, it is implausible and unappealing when it diverges greatly from utilitarianism. Nussbaum’s theory requires that enormous sums of money be devo...

By arithmeticity and superrigidity, a commensurability class of lattices in higher rank Lie group is defined by unique algebraic over number subfield $\mathbb{R}$ or $\mathbb{C}$. We prove an adelic version superrigidity which implies that two such classes define the same profinite if only groups are adelically isomorphic. discuss noteworthy consequences on rigidity questions.