Permutation-based presentations for Brin's higher-dimensional Thompson groups nVnV

The higher-dimensional Thompson groups $nV$, for $n \geq 2$, were introduced by Brin in 2005. We provide new presentations for each of these infinite simple groups. The first is an infinite presentation, analogous to the Coxeter presentation for the finite symmetric group, with generating set equal to the set of transpositions in $nV$ and reflecting the self-similar structure of $n$-dimensional Cantor space. We then exploit this infinite presentation to produce further finite presentations that are considerably smaller than those previously known.Comment: 24 pages, 2 figure

Growth of generating sets for direct powers of classical algebraic structures

For an algebraic structure A denote by d(A) the smallest size of a generating set for A, and let d(A)=(d(A),d(A2),d(A3),…), where An denotes a direct power of A. In this paper we investigate the asymptotic behaviour of the sequence d(A) when A is one of the classical structures—a group, ring, module, algebra or Lie algebra. We show that if A is finite then d(A) grows either linearly or logarithmically. In the infinite case constant growth becomes another possibility; in particular, if A is an infinite simple structure belonging to one of the above classes then d(A) is eventually constant. Where appropriate we frame our exposition within the general theory of congruence permutable varieties.Publisher PDFPeer reviewe

The infinite simple group V of Richard J. Thompson : presentations by permutations

We show that one can naturally describe elements of R. Thompson's finitely presented infinite simple group V, known by Thompson to have a presentation with four generators and fourteen relations, as products of permutations analogous to transpositions.  This perspective provides an intuitive explanation towards the simplicity of V and also perhaps indicates a reason as to why it was one of the first discovered infinite finitely presented simple groups: it is (in some basic sense) a relative of the finite alternating groups.  We find a natural infinite presentation for V as a group generated by these "transpositions," which presentation bears comparison with Dehornoy's infinite presentation and which enables us to develop two small presentations for V: a human-interpretable presentation with three generators and eight relations, and a Tietze-derived presentation with two generators and seven relations.PostprintPeer reviewe

On separability finiteness conditions in semigroups

Funding: The first author is grateful to EPSRC for financial support. The second author is grateful to the School of Mathematics and Statistics of the University of St Andrews for financial support.Taking residual finiteness as a starting point, we consider three related finiteness properties: weak subsemigroup separability, strong subsemigroup separability and complete separability. We investigate whether each of these properties is inherited by Schützenberger groups. The main result of this paper states that for a finitely generated commutative semigroup S, these three separability conditions coincide and are equivalent to every H -class of S being finite. We also provide examples to show that these properties in general differ for commutative semigroups and finitely generated semigroups. For a semigroup with finitely many H -classes, we investigate whether it has one of these properties if and only if all its Schützenberger groups have the property.Publisher PDFPeer reviewe

Automorphism groups of countable algebraically closed graphs and endomorphisms of the random graph

We establish links between countable algebraically closed graphs and the endomorphisms of the countable universal graph $R$. As a consequence we show that, for any countable graph $\Gamma$, there are uncountably many maximal subgroups of the endomorphism monoid of $R$ isomorphic to the automorphism group of $\Gamma$. Further structural information about End $R$ is established including that Aut $\Gamma$ arises in uncountably many ways as a Sch\"{u}tzenberger group. Similar results are proved for the countable universal directed graph and the countable universal bipartite graph.Comment: Minor revision following referee's comments. 27 pages, 3 figure

Automorphism groups of linearly ordered structures and endomorphisms of the ordered set ( Q ,≤) of rational numbers

We investigate the structure of the monoid of endomorphisms of the ordered set ( Q ,≤) of rational numbers. We show that for any countable linearly ordered set Ω, there are uncountably many maximal subgroups of End( Q ,≤) isomorphic to the automorphism group of Ω. We characterize those subsets X of Q that arise as a retract in ( Q ,≤) in terms of topological information concerning X. Finally, we establish that a countable group arises as the automorphism group of a countable linearly ordered set, and hence as a maximal subgroup of End( Q ,≤), if and only if it is free abelian of finite rank.PreprintPostprintPeer reviewe

IL-6 Mediated Degeneration of Forebrain GABAergic Interneurons and Cognitive Impairment in Aged Mice through Activation of Neuronal NADPH Oxidase

BACKGROUND:Multiple studies have shown that plasma levels of the pro-inflammatory cytokine interleukin-6 (IL-6) are elevated in patients with important and prevalent adverse health conditions, including atherosclerosis, diabetes, obesity, obstructive sleep apnea, hypertension, and frailty. Higher plasma levels of IL-6, in turn, increase the risk of many conditions associated with aging including age-related cognitive decline. However, the mechanisms underlying this association between IL-6 and cognitive vulnerability remain unclear. METHODS AND FINDINGS:We investigated the role of IL-6 in brain aging in young (4 mo) and aged (24 mo) wild-type C57BL6 and genetically-matched IL-6(-/-) mice, and determined that IL-6 was necessary and sufficient for increased neuronal expression of the superoxide-producing immune enzyme, NADPH-oxidase, and this was mediated by non-canonical NFkappaB signaling. Furthermore, superoxide production by NADPH-oxidase was directly responsible for age-related loss of parvalbumin (PV)-expressing GABAergic interneurons, neurons essential for normal information processing, encoding, and retrieval in hippocampus and cortex. Targeted deletion of IL-6 or elimination of superoxide by chronic treatment with a superoxide-dismutase mimetic prevented age-related loss of PV-interneurons and reversed age-related cognitive deficits on three standard tests of spatial learning and recall. CONCLUSIONS:Present results indicate that IL-6 mediates age-related loss of critical PV-expressing GABAergic interneurons through increased neuronal NADPH-oxidase-derived superoxide production, and that rescue of these interneurons preserves cognitive performance in aging mice, suggesting that elevated peripheral IL-6 levels may be directly and mechanistically linked to long-lasting cognitive deficits in even normal older individuals. Further, because PV-interneurons are also selectively affected by commonly used anesthetic agents and drugs, our findings imply that IL-6 levels may predict adverse CNS effects in older patients exposed to these compounds through specific derangements in inhibitory interneurons, and that therapies directed at lowering IL-6 may have cognitive benefits clinically

The Iceland Microcontinent and a continental Greenland-Iceland-Faroe Ridge

The breakup of Laurasia to form the Northeast Atlantic Realm was the culmination of a long period of tectonic unrest extending back to the Late Palaeozoic. Breakup was prolonged and complex and disintegrated an inhomogeneous collage of cratons sutured by cross-cutting orogens. Volcanic rifted margins formed, which are blanketed by lavas and underlain variously by magma-inflated, extended continental crust and mafic high-velocity lower crust of ambiguous and probably partly continental provenance. New rifts formed by diachronous propagation along old zones of weakness. North of the Greenland-Iceland-Faroe Ridge the newly forming rift propagated south along the Caledonian suture. South of the Greenland-Iceland-Faroe Ridge it propagated north through the North Atlantic Craton along an axis displaced ~ 150 km to the west of the northern rift. Both propagators stalled where the confluence of the Nagssugtoqidian and Caledonian orogens formed a transverse barrier. Thereafter, the ~ 400-km-wide latitudinal zone between the stalled rift tips extended in a distributed, unstable manner along multiple axes of extension that frequently migrated or jumped laterally with shearing occurring between them in diffuse transfer zones. This style of deformation continues to the present day. It is the surface expression of underlying magma-assisted stretching of ductile mid- and lower continental crust which comprises the Icelandic-type lower crust that underlies the Greenland-Iceland-Faroe Ridge. This, and probably also one or more full-crustal-thickness microcontinents incorporated in the Ridge, are capped by surface lavas. The Greenland-Iceland-Faroe Ridge thus has a similar structure to some zones of seaward-dipping reflectors. The contemporaneous melt layer corresponds to the 3–10 km thick Icelandic-type upper crust plus magma emplaced in the ~ 10–30-km-thick Icelandic-type lower crust. This model can account for seismic and gravity data that are inconsistent with a gabbroic composition for Icelandic-type lower crust, and petrological data that show no reasonable temperature or source composition could generate the full ~ 40-km thickness of Icelandic-type crust observed. Numerical modeling confirms that extension of the continental crust can continue for many tens of Myr by lower-crustal flow from beneath the adjacent continents. Petrological estimates of the maximum potential temperature of the source of Icelandic lavas are up to 1450 °C, no more than ~ 100 °C hotter than MORB source. The geochemistry is compatible with a source comprising hydrous peridotite/pyroxenite with a component of continental mid- and lower crust. The fusible petrology, high source volatile contents, and frequent formation of new rifts can account for the true ~ 15–20 km melt thickness at the moderate temperatures observed. A continuous swathe of magma-inflated continental material beneath the 1200-km-wide Greenland-Iceland-Faroe Ridge implies that full continental breakup has not yet occurred at this latitude. Ongoing tectonic instability on the Ridge is manifest in long-term tectonic disequilibrium on the adjacent rifted margins and on the Reykjanes Ridge, where southerly migrating propagators that initiate at Iceland are associated with diachronous swathes of unusually thick oceanic crust. Magmatic volumes in the NE Atlantic Realm have likely been overestimated and the concept of a monogenetic North Atlantic Igneous Province needs to be reappraised. A model of complex, piecemeal breakup controlled by pre-existing structures that produces anomalous volcanism at barriers to rift propagation and distributes continental material in the growing oceans fits other oceanic regions including the Davis Strait and the South Atlantic and West Indian oceans

Evaluating the Effects of SARS-CoV-2 Spike Mutation D614G on Transmissibility and Pathogenicity.

Global dispersal and increasing frequency of the SARS-CoV-2 spike protein variant D614G are suggestive of a selective advantage but may also be due to a random founder effect. We investigate the hypothesis for positive selection of spike D614G in the United Kingdom using more than 25,000 whole genome SARS-CoV-2 sequences. Despite the availability of a large dataset, well represented by both spike 614 variants, not all approaches showed a conclusive signal of positive selection. Population genetic analysis indicates that 614G increases in frequency relative to 614D in a manner consistent with a selective advantage. We do not find any indication that patients infected with the spike 614G variant have higher COVID-19 mortality or clinical severity, but 614G is associated with higher viral load and younger age of patients. Significant differences in growth and size of 614G phylogenetic clusters indicate a need for continued study of this variant