Abstract involutions of algebraic groups and of Kac-Moody groups

Based on the second author's thesis in this article we provide a uniform treatment of abstract involutions of algebraic groups and of Kac-Moody groups using twin buildings, RGD systems, and twisted involutions of Coxeter groups. Notably we simultaneously generalize the double coset decompositions established by Springer and by Helminck-Wang for algebraic groups and by Kac-Wang for certain Kac-Moody groups, we analyze the filtration studied by Devillers-Muhlherr in the context of arbitrary involutions, and we answer a structural question on the combinatorics of involutions of twin buildings raised by Bennett-Gramlich-Hoffman-Shpectorov

On the distortion of twin building lattices

We show that twin building lattices are undistorted in their ambient group; equivalently, the orbit map of the lattice to the product of the associated twin buildings is a quasi-isometric embedding. As a consequence, we provide an estimate of the quasi-flat rank of these lattices, which implies that there are infinitely many quasi-isometry classes of finitely presented simple groups. In an appendix, we describe how non-distortion of lattices is related to the integrability of the structural cocycle

Supersensitive Multifluorophore RNA‐FISH for Early Virus Detection and Flow‐FISH by Using Click Chemistry

The reliable detection of transcription events through the quantification of the corresponding mRNA is of paramount importance for the diagnostics of infections and diseases. The quantification and localization analysis of the transcripts of a particular gene allows disease states to be characterized more directly compared to an analysis on the transcriptome wide level. This is particularly needed for the early detection of virus infections as now required for emergent viral diseases, e. g. Covid‐19. In situ mRNA analysis, however, is a formidable challenge and currently performed with sets of single‐fluorophore‐containing oligonucleotide probes that hybridize to the mRNA in question. Often a large number of probe strands (>30) are required to get a reliable signal. The more oligonucleotide probes are used, however, the higher the potential off‐target binding effects that create background noise. Here, we used click chemistry and alkyne‐modified DNA oligonucleotides to prepare multiple‐fluorophore‐containing probes. We found that these multiple‐dye probes allow reliable detection and direct visualization of mRNA with only a very small number (5–10) of probe strands. The new method enabled the in situ detection of viral transcripts as early as 4 hours after infection