Alan D. Logan
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View article: Algebraically hyperbolic groups
Algebraically hyperbolic groups Open
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 dim…
View article: Fixed subgroups of generalised Baumslag-Solitar groups
Fixed subgroups of generalised Baumslag-Solitar groups Open
We investigate fixed subgroups of automorphisms of generalised Baumslag-Solitar (GBS) groups. Our main results are for automorphisms leaving a Bass-Serre tree invariant, under the assumption that all edge stabilisers are strictly contained…
View article: The conjugacy problem for ascending HNN‐extensions of free groups
The conjugacy problem for ascending HNN‐extensions of free groups Open
We give an algorithm to solve the Conjugacy Problem for ascending HNN‐extensions of free groups. To do this, we give algorithms to solve certain problems on dynamics of free group endomorphisms.
View article: Selective active resonance tuning for multi-mode nonlinear photonic cavities
Selective active resonance tuning for multi-mode nonlinear photonic cavities Open
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 …
View article: Post's Correspondence Problem for hyperbolic and virtually nilpotent groups
Post's Correspondence Problem for hyperbolic and virtually nilpotent groups Open
Post's Correspondence Problem (the PCP) is a classical decision problem in theoretical computer science that asks whether for pairs of free monoid morphisms there exists any non‐trivial such that . PCP for a group takes pairs of group homo…
View article: Post's correspondence problem for hyperbolic and virtually nilpotent groups
Post's correspondence problem for hyperbolic and virtually nilpotent groups Open
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 $g(x)=h(x)$…
View article: The Conjugacy Problem for ascending HNN-extensions of free groups
The Conjugacy Problem for ascending HNN-extensions of free groups Open
We give an algorithm to solve the Conjugacy Problem for ascending HNN-extensions of free groups. To do this, we give algorithms to solve certain problems on dynamics of free group endomorphisms.
View article: Triply-Resonant Sum Frequency Conversion with Gallium Phosphide Ring Resonators
Triply-Resonant Sum Frequency Conversion with Gallium Phosphide Ring Resonators Open
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 conv…
View article: The Surface Group Conjectures for groups with two generators
The Surface Group Conjectures for groups with two generators Open
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. M…
View article: The Equalizer Conjecture For The Free Group of Rank Two
The Equalizer Conjecture For The Free Group of Rank Two Open
The equalizer of a set of homomorphisms $S: F(a, b)\rightarrow F(\Delta)$ has rank at most two if S contains an injective map and is not finitely generated otherwise. This proves a strong form of Stallings’ Equalizer Conjecture for the fre…
View article: Target-wavelength-trimmed second harmonic generation with gallium\n phosphide-on-nitride ring resonators
Target-wavelength-trimmed second harmonic generation with gallium\n phosphide-on-nitride ring resonators Open
We demonstrate post-fabrication target-wavelength trimming with a gallium\nphosphide on silicon nitride integrated photonic platform using controlled\nelectron-beam exposure of hydrogen silsesquioxane cladding. A linear\nrelationship betwe…
View article: JSJ decompositions and polytopes for two-generator one-relator groups
JSJ decompositions and polytopes for two-generator one-relator groups Open
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 algorit…
View article: Fixed points and stable images of endomorphisms for the free group of\n rank two
Fixed points and stable images of endomorphisms for the free group of\n rank two Open
We give an algorithm which computes the fixed subgroup and the stable image\nfor any endomorphism of the free group of rank two $F_2$, answering for $F_2$ a\nquestion posed by Stallings in 1984 and a question of Ventura.\n
View article: The Post Correspondence Problem and Equalisers for Certain Free Group and Monoid Morphisms
The Post Correspondence Problem and Equalisers for Certain Free Group and Monoid Morphisms Open
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…