Jason Fox Manning

Geometry of pseudocharacters

If G is a group, a pseudocharacter f : G → R is a function which is “almost” a homomorphism. If G admits a nontrivial pseudocharacter f, we define the space of ends of G relative to f and show that if the space of ends is complicated enough, then G contains a nonabelian free group. We also construct a quasiaction by G on a tree whose space of ends contains the space of ends of G relative to f. This construction gives rise to examples of “exotic” quasi-actions on trees.

Measurement and localization of interface wave reflections from a buried target

It is demonstrated that seismic interface waves on the surface of a natural beach can be used to identify the position of a buried object. For this experiment, the waves were created with a sediment-coupling transducer and received on a three-element horizontal line array of triaxial geophones. The source and its coupling to the medium provided a high degree of signal repeatability, which was useful in improving signal-to-noise ratio. Reception of all three directions of particle velocity made it possible to augment conventional beamforming techniques with polarization filters to enhance interface-wave components. Reverberation in the beach was found to be large, though, and coherent background subtraction was required to isolate the component of the sound field reflected by the target. Propagation loss measurements provided comparisons of reflected signal power with predictions made previously, and the two were found to agree closely.

Residual finiteness, QCERF and fillings of hyperbolic groups

We prove that if every hyperbolic group is residually finite, then every quasi-convex subgroup of every hyperbolic group is separable. The main tool is relatively hyperbolic Dehn filling. 20E26, 20F67, 20F65 A group G is residually finite (or RF) if for every g2 GXf1g, there is some finite group F and an epimorphism W G! F so that . g/⁄ 1. In more sophisticated language G is RF if and only if the trivial subgroup is closed in the profinite topology on G . If H < G , then H is separable if for every g2 GXH , there is some finite group F and an epimorphism W G! F so that . g/O. H/. Equivalently, the subgroup H is separable in G if it is closed in the profinite topology on G . If every finitely generated subgroup of G is separable, G is said to be LERF or subgroup separable. If G is hyperbolic, and every quasi-convex subgroup of G is separable, we say that G is QCERF. In this paper, we show that if every hyperbolic group is RF, then every hyperbolic group is QCERF.

Separation of relatively quasiconvex subgroups

We show that if all hyperbolic groups are residually finite, these statements follow: In relatively hyperbolic groups with peripheral structures consisting of finitely generated nilpotent subgroups, relatively quasiconvex subgroups are separable; geometrically finite subgroups of nonuniform lattices in rank one symmetric spaces are separable; Kleinian groups are subgroup separable. We also show that LERF for finite volume hyperbolic 3-manifolds would follow from LERF for closed hyperbolic 3-manifolds. We prove these facts by reducing, via combination and filling theorems, the separability of a relatively quasiconvex subgroup of a relatively hyperbolic group G to that of a quasiconvex subgroup of a hyperbolic quotient N G. A result of Agol, Groves, and Manning is then applied.

QUASI-ACTIONS ON TREES AND PROPERTY (QFA)

We prove some general results about quasi-actions on trees and define Property (QFA), which is analogous to Serres Property (FA), but in the coarse setting. This property is shown to hold for a class of groups, including SL(n, Z) for n ≥ 3. We also give a way of thinking about Property (QFA) by breaking it down into statements about particular classes of trees.

Fillings, finite generation and direct limits of relatively hyperbolic groups

We examine the relationship between finitely and infinitely generated relatively hyperbolic groups. We observe that direct limits of relatively hyperbolic groups are in fact direct limits of finitely generated relatively hyperbolic groups. We combine this observation with known results to prove the Strong Novikov Conjecture for some exotic groups constructed by Osin.

Virtual geometricity is rare

We present the results of computer experiments suggesting that the probability that a random multiword in a free group is virtually geometric decays to zero exponentially quickly in the length of the multiword. We then prove this fact.

Actions of certain arithmetic groups on Gromov hyperbolic spaces

We study the variety of actions of a fixed (Chevalley) group on arbitrary geodesic, Gromov hyperbolic spaces. In high rank we obtain a complete classification. In rank one, we obtain some partial results and give a conjectural picture.

Ray theory in mode models: Demonstration of high‐frequency collective‐mode interference effects

A number of acoustic reflection and refraction effects, normally associated with ray theory, are used here as tests of a normal mode model in the high‐frequency limit, where the modes become dense in the complex wave‐number plane. The ray effects, which are associated with saddle points or branch line integrals in the wave‐number integration method, do not appear as characteristics of any single mode in the modal solution. Rather, they emerge as collective effects from the coherent addition of large numbers of modes. The effects demonstrated include beam splitting at the Brewster angle for a fluid–fluid interface and at the Rayleigh angle for a fluid–solid interface, beam displacement near the critical angle for total reflection, and beam spreading due to a fluid–solid lateral wave. Gaussian beams are modeled efficiently using the ORCA normal mode model, by assigning complex values to the source coordinates. This technique extends, in a full mode solution, methods that have previously been implemented for...

Generalized triangle groups, expanders, and a problem of Agol and Wise

Answering a question asked by Agol and Wise, we show that a desired stronger form of Wises malnormal special quotient theorem does not hold. The counterexamples are generalizations of triangle groups, built using the Ramanujan graphs constructed by Lubotzky--Phillips--Sarnak.