Kanai proved powerful results on the stability under quasi-isometries of numerous global properties (including Liouville property) between Riemannian manifolds of bounded geometry. Since his work focuses more on the generality of the spaces considered than on the two-dimensional geometry, Kanai's hypotheses in many cases are not satisfied in the context of Riemann surfaces endowed with the Poincaré metric. In this work we fill that gap for the Liouville property, by proving its stability by quasi-isometries for every Riemann surface (and even Riemannian surfaces with pinched negative curvature). Also, a key result characterizes Riemannian surfaces which are quasi-isometric to $\mathbb {R}$.

Kant’s theory of friendship is crucial in defending his ethics against the longstanding charge of emotional detachment. But his theory of friendship is vulnerable to this charge too: the Kantian sage can appear to reject sympathetic suffering when she cannot help a suffering friend. I argue that Kant is committed to the view that both sages and ordinary people must suffer in sympathy with friends even when they cannot help, because sympathy is necessary to fulfill the imperfect duty to adopt others’ merely permissible ends (MPEs), and we ought to take friends’ MPEs as our own. MPEs are individuated in terms of concepts which include marks of the first person, and no marks of law other than permissibility. To adopt ends of others individuated in terms of such concepts rather than merely promote them as means to different ends, those concepts must engage with one’s feelings in a way that requires sympathy.

This paper explores the effect that conformity to the rule of law has on the ends which might legitimately be pursued within a legal system. The neat distinction between formal and substantive conceptions of the rule of law will be challenged: even apparently formal conceptions necessarily affect the content of law and necessarily entail the protection of certain fundamental rights. What remains of the formal/substantive dichotomy is, in fact, a distinction between conceptions of the rule of law which guarantee the substantive justice of each and every law and those which entail some commitment to basic requirements of justice while nevertheless leaving room for unjust laws. Ultimately, the only significant distinction between competing theories of the rule of law concerns the nature of the connection between legality and justice, not whether there is any such connection at all.

There are several senses in which Kant’s moral law is independent of sensibility. This paper is devoted mainly to Kant’s account of ‘physical conditions independence’, or the idea that the moral law can compel us to pursue ends that might be impossible to realize empirically. Since this idea has received little attention from commentators, this paper addresses both its textual basis in Kant’s writings and its overall philosophical viability.

Kant famously distinguishes between the categorical imperative (CI) – the fundamental principle of morality – and hypothetical imperatives (HIs), which are instrumental norms. On the standard reading, Kant subscribes to the ‘disjunctive reading’ of HIs, which takes HIs to be consistency requirements that bind agents in exactly the same way whether or not agents are subject to CI and whether or not they conform their choices to CI. I argue that this reading cannot be squared with Kant's account of an agent's disposition, in particular his claim that cognition of CI is a necessary condition of willing a maxim. I further argue that Kant could not accept an account of HIs as consistency requirements. Finally, I outline Kant's conception of HIs as non-disjunctive requirements that arise when and only when agents will permissible ends. This account can help recapture Kant's conception of the unity of rational norms.

The purpose of this article is to draw on Le Grand's (2007) model of service provision of ends and means (trust, targets, voice and choice) to critique the accepted frameworks for conceptualising the chronology of equal opportunities and diversity (EO&D) in the UK. We do this by reviewing the attempts to provide a chronological analysis before outlining and applying Le Grand's (2007) model. We find that the ‘eras’ of the chronologies give way to a much more complex and fluid picture. Moreover, focusing on ends and means highlights some major issues in the development of EO&D policy that needs to be addressed.

We give a graph-theoretic definition for the number of ends of Cayley digraphs for finitely generated semigroups and monoids. For semigroups and monoids, left Cayley digraphs can be very different from right Cayley digraphs. In either case, the number of ends for the Cayley digraph does not depend upon which finite set of generators is used for the semigroup or monoid. For natural numbers m and n, we exhibit finitely generated monoids for which the left Cayley digraphs have m ends while the right Cayley digraphs have n ends. For direct products and for many other semidirect products of a pair of finitely generated infinite monoids, the right Cayley digraph of the semidirect product has only one end. A finitely generated subsemigroup of a free semigroup has either one end or else has infinitely many ends.