Let CS denote the class of groups isomorphic to the fundamental group of two closed hyperbolic surfaces identified along an essential simple closed curve in each. We construct a bi-Lipschitz map between the universal covers of these spaces equipped with a CAT(−1) metric, proving all groups in CS are quasi-isometric. The class CS has infinitely many abstract commensurability classes, which we ch...

The aim of this paper is to shed some light on the use of fuzzy measures and integrals as aggregation operators in multicriteria decision making. These techniques have been widely used on an ad hoc basis, but with no ax-iomatization. It is possible to obtain preference representation theorems in multicriteria decision making problems, relying on a formal parallelism between decision under uncer...

There does not exist a Borel way of selecting an isomorphism class within each commensurability class of finitely generated groups.

The commensurability condition is applied to determine the hierarchy of fractional fillings of Landau levels in monolayer and in bilayer graphene. The filling rates for fractional quantum Hall effect (FQHE) in graphene are found in the first three Landau levels in one-to-one agreement with the experimental data. The presence of even denominator filling fractions in the hierarchy for FQHE in bil...

This essay offers an interpretation of Aristotle's remarks on the commensurability of goods in Book V of the Nicomachean Ethics. It explores the term ‘by hypothesis’ (ἐξ ὑποθέσεως) which Aristotle uses to describe the institution of currency through which commensurability is established. The term implies that Aristotle conceives the origins of currency to lie in a conscious act of stipulation r...

In this paper we prove that if MK is the complement of a non-fibered twist knot K in S, then MK is not commensurable to a fibered knot complement in a Z/2Z-homology sphere. To prove this result we derive a recursive description of the character variety of twist knots and then prove that a commensurability criterion developed by D. Calegari and N. Dunfield is satisfied for these varieties. In ad...

This paper is based on three lectures given by the author at the workshop, “Hyperbolic geometry and arithmetic: a crossview” held at The Université Paul Sabatier, Toulouse in November 2012. The goal of the lectures was to describe recent work on the extent to which various geometric and analytical properties of hyperbolic 3-manifolds determine the commensurability class of such manifolds. This ...