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Alan D. Logan

Publications and source records attributed to Alan D. Logan.

At least 19 recordsLinked to original sources
arXiv2025

Algebraically hyperbolic groups

We initiate the study of torsion-free algebraically hyperbolic groups; these groups generalise torsion-free hyperbolic groups and are intricately related to groups with no Baumslag--Solitar subgroups. Indeed, for groups of cohomological dimension $2$ we prove that algebraic hyperbolicity is equivalent to containing no Baumslag--Solitar subgroups. This links algebraically hyperbolic groups to two famous questions of Gromov; recent work has shown these questions to have negative answers in general, but they remain open for groups of cohomological dimension $2$. We also prove that algebraically hyperbolic groups are CSA, and so have canonical abelian JSJ-decompositions. In the two-generated case we give a precise description of the form of these decompositions.

math.GR↗
arXiv2023

Selective active resonance tuning for multi-mode nonlinear photonic cavities

Resonant enhancement of nonlinear photonic processes is critical for the scalability of applications such as long-distance entanglement generation. To implement nonlinear resonant enhancement, multiple resonator modes must be individually tuned onto a precise set of process wavelengths, which requires multiple linearly-independent tuning methods. Using coupled auxiliary resonators to indirectly tune modes in a multi-resonant nonlinear cavity is particularly attractive because it allows the extension of a single physical tuning mechanism, such as thermal tuning, to provide the required independent controls. Here we model and simulate the performance and tradeoffs of a coupled-resonator tuning scheme which uses auxiliary resonators to tune specific modes of a multi-resonant nonlinear process. Our analysis determines the tuning bandwidth for steady-state mode field intensity can significantly exceed the inter-cavity coupling rate if the total quality factor of the auxiliary resonator is higher than the multi-mode main resonator. Consequently, over-coupling a nonlinear resonator mode to improve the maximum efficiency of a frequency conversion process will simultaneously expand the auxiliary resonator tuning bandwidth for that mode, indicating a natural compatibility with this tuning scheme. We apply the model to an existing small-diameter triply-resonant ring resonator design and find that a tuning bandwidth of 136 GHz ~ 1.1 nm can be attained for a mode in the telecom band while limiting excess scattering losses to a quality factor of 10^6. Such range would span the distribution of inhomogeneously broadened quantum emitter ensembles as well as resonator fabrication variations, indicating the potential for the auxiliary resonators to enable not only low-loss telecom conversion but also the generation of indistinguishable photons in a quantum network.

physics.optics↗
arXiv2023

Post's correspondence problem for hyperbolic and virtually nilpotent groups

Post's Correspondence Problem (the PCP) is a classical decision problem in theoretical computer science that asks whether for pairs of free monoid morphisms $g, h\colonΣ^*\toΔ^*$ there exists any non-trivial $x\inΣ^*$ such that $g(x)=h(x)$. Post's Correspondence Problem for a group $Γ$ takes pairs of group homomorphisms $g, h\colon F(Σ)\to Γ$ instead, and similarly asks whether there exists an $x$ such that $g(x)=h(x)$ holds for non-elementary reasons. The restrictions imposed on $x$ in order to get non-elementary solutions lead to several interpretations of the problem; we consider the natural restriction asking that $x \notin \ker(g) \cap \ker(h)$ and prove that the resulting interpretation of the PCP is undecidable for arbitrary hyperbolic $Γ$, but decidable when $Γ$ is virtually nilpotent. We also study this problem for group constructions such as subgroups, direct products and finite extensions. This problem is equivalent to an interpretation due to Myasnikov, Nikolev and Ushakov when one map is injective.

math.GR↗
arXiv2022

Triply-Resonant Sum Frequency Conversion with Gallium Phosphide Ring Resonators

We demonstrate quasi-phase matched, triply-resonant sum frequency conversion in 10.6-um-diameter integrated gallium phosphide ring resonators. A small-signal, waveguide-to-waveguide power conversion efficiency of 8%/mW is measured for conversion from telecom (1536 nm) and near infrared (1117 nm) to visible (647 nm) wavelengths with an absolute power conversion efficiency of 6.3% measured at saturation pump power. For the complementary difference frequency generation process, a single photon conversion efficiency of 7.2%/mW from visible to telecom is projected for resonators with optimized coupling. Efficient conversion from visible to telecom will facilitate long-distance transmission of spin-entangled photons from solid-state emitters such as the diamond NV center, allowing long-distance entanglement for quantum networks.

physics.optics↗
arXiv2022

The Surface Group Conjectures for groups with two generators

The Surface Group Conjectures are statements about recognising surface groups among one-relator groups, using either the structure of their finite-index subgroups, or all subgroups. We resolve these conjectures in the two generator case. More generally, we prove that every two-generator one-relator group with every infinite-index subgroup free is itself either free or a surface group.

math.GR↗
arXiv2021

The equalizer conjecture for the free group of rank two

The equaliser of a set of homomorphisms $S: F(a, b)\rightarrow F(Δ)$ has rank at most two if $S$ contains an injective map, and is not finitely generated otherwise. This proves a strong form of Stallings' Equaliser Conjecture for the free group of rank two. Results are also obtained for pairs of homomorphisms $g, h:F(Σ)\rightarrow F(Δ)$ when the images are inert in, or retracts of, $F(Δ)$.

math.GR↗
arXiv2021

Target-wavelength-trimmed second harmonic generation with gallium phosphide-on-nitride ring resonators

We demonstrate post-fabrication target-wavelength trimming with a gallium phosphide on silicon nitride integrated photonic platform using controlled electron-beam exposure of hydrogen silsesquioxane cladding. A linear relationship between the electron-beam exposure dose and resonant wavelength red-shift enables deterministic, individual trimming of multiple devices on the same chip to within 30 pm of a single target wavelength. Second harmonic generation from telecom to near infrared at a target wavelength is shown in multiple devices with quality factors on the order of $10^4$. Post-fabrication tuning is an essential tool for targeted wavelength applications including quantum frequency conversion

physics.optics↗
arXiv2021

Variations on the Post Correspondence Problem for free groups

The Post Correspondence Problem is a classical decision problem about equalisers of free monoid homomorphisms. We prove connections between several variations of this classical problem, but in the setting of free groups and free group homomorphisms. Among other results, and working under certain injectivity assumptions, we prove that computing the rank of the equaliser of a pair of free group homomorphisms can be applied to computing a basis of this equaliser, and also to solve the "generalised" Post Correspondence Problem for free groups.

math.GR↗
arXiv2021

JSJ decompositions and polytopes for two-generator one-relator groups

We provide a direct connection between the Z_{max} (or essential) JSJ decomposition and the Friedl--Tillmann polytope of a hyperbolic two-generator one-relator group with abelianisation of rank $2$. We deduce various structural and algorithmic properties, like the existence of a quadratic-time algorithm computing the Z_{max}-JSJ decomposition of such groups.

math.GR↗
arXiv2020

The Post Correspondence Problem and equalisers for certain free group and monoid morphisms

A marked free monoid morphism is a morphism for which the image of each generator starts with a different letter, and immersions are the analogous maps in free groups. We show that the (simultaneous) PCP is decidable for immersions of free groups, and provide an algorithm to compute bases for the sets, called equalisers, on which the immersions take the same values. We also answer a question of Stallings about the rank of the equaliser. Analogous results are proven for marked morphisms of free monoids.

math.GR↗
arXiv2019

Every group is the outer automorphism group of an HNN-extension of a fixed triangle group

Fix an equilateral triangle group $T_i=\langle a, b; a^i, b^i, (ab)^i\rangle$ with $i\geq6$ arbitrary. Our main result is: for every presentation $\mathcal{P}$ of every countable group $Q$ there exists an HNN-extension $T_{\mathcal{P}}$ of $T_i$ such that $\operatorname{Out}(T_{\mathcal{P}})\cong Q$. We construct the HNN-extensions explicitly, and examples are given. The class of groups constructed have nice categorical and residual properties. In order to prove our main result we give a method for recognising malnormal subgroups of small cancellation groups, and we introduce the concept of "malcharacteristic" subgroups.

math.GR↗
arXiv2018

The Bass-Jiang group for automorphism-induced HNN-extensions

We use the Bass--Jiang group for automorphism-induced HNN-extensions to build a framework for the construction of tractable groups with pathological outer automorphism groups. We apply this framework to a strong form of a question of Bumagin--Wise on the outer automorphism groups of finitely presented, residually finite groups.

math.GR↗
arXiv2018

400%/W second harmonic conversion efficiency in $\mathrm{14 μm}$-diameter gallium phosphide-on-oxide resonators

Second harmonic conversion from 1550~nm to 775~nm with an efficiency of 400% W$^{-1}$ is demonstrated in a gallium phosphide (GaP) on oxide integrated photonic platform. The platform consists of doubly-resonant, phase-matched ring resonators with quality factors $Q \sim 10^4$, low mode volumes $V \sim 30 (λ/n)^3$, and high nonlinear mode overlaps. Measurements and simulations indicate that conversion efficiencies can be increased by a factor of 20 by improving the waveguide-cavity coupling to achieve critical coupling in current devices.

physics.app-ph↗
arXiv2018

The residual finiteness of (hyperbolic) automorphism-induced HNN-extensions

We classify finitely generated, residually finite automorphism-induced HNN-extensions in terms of the residual separability of a single associated subgroup. This classification provides a method to construct automorphism-induced HNN-extensions which are not residually finite. We prove that this method can never yield a "new" counter-example to Gromov's conjecture on the residual finiteness of hyperbolic groups.

math.GR↗
arXiv2016

The JSJ-decompositions of one-relator groups with torsion

In this paper we use JSJ-decompositions to formalise a folk conjecture recorded by Pride on the structure of one-relator groups with torsion. We prove a slightly weaker version of the conjecture, which implies that the structure of one-relator groups with torsion closely resemble the structure of torsion-free hyperbolic groups.

math.GR↗
arXiv2015

The Outer Automorphism Groups of Two-Generator One-Relator Groups with Torsion

The main result of this paper is a complete classification of the outer automorphism groups of two-generator, one-relator groups with torsion. To this classification we apply recent algorithmic results of Dahmani--Guirardel, which yields an algorithm to compute the isomorphism class of the outer automorphism group of a given two-generator, one-relator group with torsion.

math.GR↗