Actions of arithmetic groups on homology spheres and acyclic homology manifolds

We establish lower bounds on the dimensions in which arithmetic groups with torsion can act on acyclic manifolds and homology spheres. The bounds rely on the existence of elementary p-groups in the groups concerned. In some cases, including Sp(2n,Z), the bounds we obtain are sharp: if X is a generalized Z/3-homology sphere of dimension less than 2n-1 or a Z/3-acyclic Z/3-homology manifold of dimension less than 2n, and if n \geq 3, then any action of Sp(2n,Z) by homeomorphisms on X is trivial; if n = 2, then every action of Sp(2n,Z) on X factors through the abelianization of Sp(4,Z), which is Z/2.Comment: Final version, to appear in Math Zeitschrif

The symmetries of outer space

Published versio

Tree-irreducible automorphisms of free groups

We introduce a new class of automorphisms $\varphi$ of the non-abelian free group $F_N$ of finite rank $N \geq 2$ which contains all iwips (= fully irreducible automorphisms), but also any automorphism induced by a pseudo-Anosov homeomorphism of a surface with arbitrary many boundary components. More generally, there may be subgroups of $F_N$ of rank $\geq 2$ on which $\varphi$ restricts to the identity. We prove some basic facts about such {\em tree-irreducible} automorphisms, and show that, together with Dehn twist automorphisms, they are the natural basic building blocks from which any automorphism of \FN can be constructed in a train track set-up. We then show: {\bf Theorem:} {\it Every tree-irreducible automorphism of $F_N$ has induced North-South dynamics on the Thurston compactification $\bar{\rm CV}_N$ of Outer space.} Finally, we define a "blow-up" construction on the vertices of a train track map, which, starting from iwips, produces tree-irreducible automorphisms which in general are not iwip

Dimension of the Torelli group for Out(F_n)

Let T_n be the kernel of the natural map from Out(F_n) to GL(n,Z). We use combinatorial Morse theory to prove that T_n has an Eilenberg-MacLane space which is (2n-4)-dimensional and that H_{2n-4}(T_n,Z) is not finitely generated (n at least 3). In particular, this recovers the result of Krstic-McCool that T_3 is not finitely presented. We also give a new proof of the fact, due to Magnus, that T_n is finitely generated.Comment: 27 pages, 9 figure

Classical and quantum mechanical plane switching in CO2

Classical plane switching takes place in systems with a pronounced 1:2 resonance, where the degree of freedom with lowest frequency is doubly-degenerate. Under appropriate conditions, one observes a periodic and abrupt precession of the plane in which the doubly-degenerate motion takes place. In this article, we show that quantum plane switching exists in CO2 : Based on our analytical solutions of the classical Hamilton's equations of motion, we describe the dependence on vibrational angular momentum and energy of the frequency of switches and the plane switching angle. Using these results, we find optimal initial wave packet conditions for CO2 and show, through quantum mechanical propagation, that such a wave packet indeed displays plane switching at energies around 10000 cm-1 above the ground state on time scales of about 100 fs.Comment: accepted for publication in the Journal of Chemical Physic

Stability of the homology of the moduli spaces of Riemann surfaces with spin structure

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/46234/1/208_2005_Article_BF01446896.pd

Intersection form, laminations and currents on free groups

Let $F_N$ be a free group of rank $N\ge 2$, let $\mu$ be a geodesic current on $F_N$ and let $T$ be an $\mathbb R$-tree with a very small isometric action of $F_N$. We prove that the geometric intersection number $$ is equal to zero if and only if the support of $\mu$ is contained in the dual algebraic lamination $L^2(T)$ of $T$. Applying this result, we obtain a generalization of a theorem of Francaviglia regarding length spectrum compactness for currents with full support. As another application, we define the notion of a \emph{filling} element in $F_N$ and prove that filling elements are "nearly generic" in $F_N$. We also apply our results to the notion of \emph{bounded translation equivalence} in free groups.Comment: revised version, to appear in GAF

Clinical and virological characteristics of hospitalised COVID-19 patients in a German tertiary care centre during the first wave of the SARS-CoV-2 pandemic: a prospective observational study

Purpose: Adequate patient allocation is pivotal for optimal resource management in strained healthcare systems, and requires detailed knowledge of clinical and virological disease trajectories. The purpose of this work was to identify risk factors associated with need for invasive mechanical ventilation (IMV), to analyse viral kinetics in patients with and without IMV and to provide a comprehensive description of clinical course. Methods: A cohort of 168 hospitalised adult COVID-19 patients enrolled in a prospective observational study at a large European tertiary care centre was analysed. Results: Forty-four per cent (71/161) of patients required invasive mechanical ventilation (IMV). Shorter duration of symptoms before admission (aOR 1.22 per day less, 95% CI 1.10-1.37, p < 0.01) and history of hypertension (aOR 5.55, 95% CI 2.00-16.82, p < 0.01) were associated with need for IMV. Patients on IMV had higher maximal concentrations, slower decline rates, and longer shedding of SARS-CoV-2 than non-IMV patients (33 days, IQR 26-46.75, vs 18 days, IQR 16-46.75, respectively, p < 0.01). Median duration of hospitalisation was 9 days (IQR 6-15.5) for non-IMV and 49.5 days (IQR 36.8-82.5) for IMV patients. Conclusions: Our results indicate a short duration of symptoms before admission as a risk factor for severe disease that merits further investigation and different viral load kinetics in severely affected patients. Median duration of hospitalisation of IMV patients was longer than described for acute respiratory distress syndrome unrelated to COVID-19