Our main result asserts that for any given numbers C and D the class of simply connected closed smooth manifolds of dimension m < 7 which admit a Riemannian metric with sectional curvature bounded in absolute value by |K| ≤ C and diameter uniformly bounded from above by D contains only finitely many diffeomorphism types. Thus in these dimensions the lower positive bound on volume in Cheeger’s F...

We consider discrete one-dimensional Schrödinger operators with strictly ergodic, aperiodic potentials taking finitely many values. The well-known tendency of these operators to have purely singular continuous spectrum of zero Lebesgue measure is further elucidated. We study stability of singular continuity with respect to local perturbations. Moreover, we provide a unified approach to both the...

There is a forgetful map from the mapping class group of a punctured surface to that of the surface with one fewer puncture. We relate the action of the kernel on the curve complex to a family of actions on trees. Together with a geometric compactness argument we further prove that finitely generated purely pseudo-Anosov subgroups of the kernel are convex cocompact in the sense of Farb and Mosh...

In this paper, we explore semantics for comparative epistemic modals that avoid the entailment problems shown by Yalcin (2006, 2009, 2010) to result from Kratzer’s (1991) semantics. In contrast to the alternative semantics presented by Yalcin and Lassiter (2010, 2011) based on finitely additive measures, we introduce semantics based on qualitatively additive measures, as well as semantics based...

Let λ̄ be any atomless and countably additive probability measure on the product space {0,1}N with the usual σ -algebra. Then there is a purely finitely additive probability measure λ on the power set of a countable subset T ⊂ T̄ such that Lp(λ̄) can be isometrically isomorphically embedded as a closed subspace of Lp(λ). The embedding is strict. It is also ‘canonical,’ in the sense that it maps si...

Largeness, SQ-universality, and the existence of free subgroups of rank 2 are measures of the complexity of a finitely presented group. We obtain conditions under which a cyclically presented group possesses one or more of these properties. We apply our results to a class of groups introduced by Prishchepov which contain, amongst others, the various generalizations of Fibonacci groups introduce...

A group is said to have the R∞ property if every automorphism has an infinite number of twisted conjugacy classes. We study the question whether G has the R∞ property when G is a finitely generated torsion-free nilpotent group. As a consequence, we show that for every positive integer n ≥ 5, there is a compact nilmanifold of dimension n on which every homeomorphism is isotopic to a fixed point ...

We consider discrete one-dimensional Schrödinger operators with strictly ergodic, aperiodic potentials taking finitely many values. The well-known tendency of these operators to have purely singular continuous spectrum of zero Lebesgue measure is further elucidated. We provide a unified approach to both the study of the spectral type as well as the measure of the spectrum as a set. We apply thi...

Here ~9(x):p(x; wl,w3) denotes the elliptic Weierstrass function with fundamental periods 2wl and 2w3 (Im(w3/Wl)#0). In the special case where Wl is real and w3 is purely imaginary, the potential q(x) in (1.1) is real-valued and Ince's striking result [51], in modern spectral-theoretic terminology, yields that the spectrum of the unique self-adjoint operator associated with the differential exp...