arXiv subjects

Gaven Martin

Publications and source records attributed to Gaven Martin.

25 records · Page 2Linked to original sources

arXiv2020

Nondiscrete parabolic characters of the free group $F_2$: supergroup density and Nielsen classes in the complement of the Riley slice

A parabolic representation of the free group $F_2$ is one in which the images of both generators are parabolic elements of $PSL(2,\IC)$. The Riley slice is a closed subset ${\cal R}\subset \IC$ which is a model for the parabolic, discrete and faithful characters of $F_2$. The complement of the Riley slice is a bounded Jordan domain within which there are isolated points, accumulating only at the boundary, corresponding to parabolic discrete and faithful representations of rigid subgroups of $PSL(2,\IC)$. Recent work of Aimi, Akiyoshi, Lee, Oshika, Parker, Lee, Sakai, Sakuma \& Yoshida, have topologically identified all these groups. Here we give the first identified substantive properties of the nondiscrete representations and prove a supergroup density theorem: given any irreducible parabolic representation $ρ_*:F_2\to PSL(2,\IC)$ whatsoever, any non-discrete parabolic representation $ρ_0$ has an arbitrarily small perturbation $ρ_ε$ so that $ρ_ε(F_2)$ contains a conjugate of $ρ_*(F_2)$ as a proper subgroup. This implies that if $Γ_*$ is any nonelementary group generated by two parabolic elements (discrete or otherwise) and $γ_0$ is any point in the complement of the Riley slice, then in any neighbourhood of $γ$ there is a point corresponding to a nonelementary group generated by two parabolics with a conjugate of $Γ_*$ as a proper subgroup. Using these ideas we then show that there are nondiscrete parabolic representations with an arbitrarily large number of distinct Nielsen classes of parabolic generators.

math.CV↗

arXiv2019

Non-variational extrema of exponential Teichmüller spaces

The exponential Teichmüller spaces $E_p$, $0\leq p \leq \infty$, interpolate between the classical Teichmüller space ($p=\infty$) and the space of harmonic diffeomorphisms $(p=0)$. In this article we prove the existence of non-variational critical points for the associated functional: mappings $f$ of the disk whose distortion is $p$-exponentially integrable, $0<p<\infty$, yet for {\em any} diffeomorphism $g(z)$ of $\ID$ with $g|\partial\ID=identity$ and $g\neq identity$ we have $f\circ g$ is not of $p$-exponentially integrable distortion.

math.CV↗

arXiv2018

A resonance interaction of seismogravitational modes on tectonic plates

This paper discusses resonance effects to advance a classical earthquake model, namely the celebrated M8 global test algorithm. This algorithmgives high confidence levels for prediction of Time Intervals of Increased Probability (TIP) of an earthquake. It is based on observation that almost 80\% of earthquakes occur due to the stress accumulated from previous earthquakes at the location and stored in form of displacements against gravity and static elastic deformations of the plates. Nevertheless the M8 global test algorithm fails to predict some powerful earthquakes. In this paper we suggest the additional possibility of considering the dynamical storage of the elastic energy on the tectonic plates due to resonance beats of seismo-gravitational oscillations (SGO) modes of the plates. We make sure that the tangential compression in the middle plane of an "active zone" of a tectonic plate may tune its SGO modes to the resonance condition of coincidence the frequencies of the corresponding localized modes with the delocalized SGO modes of the complement. We also consider the beats arising between the modes under a small perturbations of the plates, and, assuming that the discord between the perturbed and unperturbed resonance modes is strongly dominated by the discord between the non-resonance modes estimate the energy transfer coefficient.

math.SP↗

arXiv2018

Quasiregular families bounded in $L^p$ and elliptic estimates

We prove that a family of quasiregular mappings of a domain $Ω$ which are uniformly bounded in $L^p$ for some $p>0$ form a normal family. From this we show how an elliptic estimate on a functional differences implies all directional derivatives, and thus the complex gradient to be quasiregular. Consequently the function enjoys much higher regularity than apriori assumptions suggest.

math.CV↗

arXiv2018

Random Kleinian Groups, II : Two parabolic generators

In earlier work we introduced geometrically natural probability measures on the group of all Möbius transformations in order to study "random" groups of Möbius transformations, random surfaces, and in particular random two-generator groups, that is groups where the generators are selected randomly, with a view to estimating the likely-hood that such groups are discrete and then to make calculations of the expectation of their associated parameters, geometry and topology. In this paper we continue that study and identify the precise probability that a Fuchsian group generated by two parabolic Möbius transformations is discrete, and give estimates for the case of Kleinian groups generated by a pair of random parabolic elements which we support with a computational investigation into of the Riley slice as identified by Bowditch's condition, and establish rigorous bounds.

math.CV↗

arXiv2017

Random Kleinian Groups I: Random Fuchsian Groups

We introduce a geometrically natural probability measure on the group of all Möbius transformations of the circle. Our aim is to study "random" groups of Möbius transformations, and in particular random two-generator groups. By this we mean groups where the generators are selected randomly. The probability measure in effect establishes an isomorphism between random $n$-generators groups and collections of $n$ random pairs of arcs on the circle. Our aim is to estimate the likely-hood that such a random group is discrete, calculate the expectation of their associated parameters, geometry and topology, and to test the effectiveness of tests for discreteness such as Jørgensen's inequality.

math.CV↗

arXiv2015

The (p, q)-arithmetic hyperbolic lattices; p, q greater than or equal to 6

We prove there are exactly 16 arithmetic lattices of hyperbolic 3-space which are generated by two elements of finite orders p and q with p,q at least six. We also verify a conjecture of H.M. Hilden, M.T. Lozano, and J.M. Montesinos concerning the orders of the singular sets of arithmetic orbifold Dehn surgeries on two bridge knot and link complements.

math.GT↗