We consider the following conjecture. Suppose that G is a non-free non-cyclic onerelator group such that each subgroup of finite index is again a one-relator group and each subgroup of infinite index is a free group. Must G be a surface group? We show that if G is a freely indecomposable fully residually free group and satisfies the property that every subgroup of infinite index is free then G ...

This is an expository article based on the author’s lecture delivered at the conference Lie Theory and Its Applications in March 2011, UCSD. We discuss various notions of positivity and their relations with the study of the Ricci flow, including a proof of the assertion, due to Wolfson and the author, that the Ricci flow preserves the positivity of the complex sectional curvature. We discuss th...

Let / be a C°° Anosov diffeomorphism of a compact man-ifold M, preserving a smooth measure. If / satisfies the |-pinching assumption defined below, it must preserve a continuous affine connection for which the leaves of the Anosov foliations are totally geodesic, geodesically complete, and flat (its tangential curvature is defined along individual leaves). If this connection, which is the uniqu...

We discuss the memristive fingerprints of the hybrid Cassie-Mayr model of electric arcs. In particular, it is shown that (i) the voltage-current characteristic of the model has the pinched hysteresis nature, (ii) the voltage and current zero crossings occur at the same instants, and (iii) when the frequency f of the power supply increases, the voltage-current pinched hysteresis characteristic t...

We show that a noncompact manifold with bounded sectional curvature, whose ends are sufficiently collapsed, has a finite dimensional space of square-integrable harmonic forms. In the special case of a finite-volume manifold with pinched negative sectional curvature, we show that the essential spectrum of the p-form Laplacian is the union of the essential spectra of a collection of ordinary diff...