Reflections in abstract Coxeter groups

Let $W$ be a Coxeter group and $r\in W$ a reflection. If the group of order 2 generated by $r$ is the intersection of all the maximal finite subgroups of $W$ that contain it, then any isomorphism from $W$ to a Coxeter group $W'$ must take $r$ to a reflection in $W'$. The aim of this paper is to show how to determine, by inspection of the Coxeter graph, the intersection of the maximal finite sugroups containing $r$. In particular we show that the condition above is satisfied whenever $W$ is infinite and irreducible, and has the property that all rank two parabolic subgroups are finite. So in this case all isomorphisms map reflections to reflections.Comment: 25 pages, 0 figure

The Center Conjecture for spherical buildings of types F4 and E6

We prove that a convex subcomplex of a spherical building of type F4 or E6 is a subbuilding or the automorphisms of the subcomplex fix a point on it. Our approach is differential-geometric and based on the theory of metric spaces with curvature bounded above. We use these techniques also to give another proof of the same result for the spherical buildings of classical type.Comment: 34 pages. An intrinsic version of the results has been added. Proof of the Center Conjecture for spherical buildings of classical types added. More material on Coxeter complexe

An outline of polar spaces: basics and advances

This paper is an extended version of a series of lectures on polar spaces given during the workshop and conference 'Groups and Geometries', held at the Indian Statistical Institute in Bangalore in December 2012. The aim of this paper is to give an overview of the theory of polar spaces focusing on some research topics related to polar spaces. We survey the fundamental results about polar spaces starting from classical polar spaces. Then we introduce and report on the state of the art on the following research topics: polar spaces of infinite rank, embedding polar spaces in groups and projective embeddings of dual polar spaces

Direct Evidence for Natural Transmission of Small-Ruminant Lentiviruses of Subtype A4 from Goats to Sheep and Vice Versa

Small-ruminant lentiviruses (SRLV), which include the caprine arthritis-encephalitis and the maedi-visna virus, cause persistent inflammatory infections in goats and sheep. SRLV are mainly transmitted from mother to offspring through milk. Transmission after prolonged contact between adult animals has also been observed. The observation that certain SRLV subtypes are found in both goats and sheep suggests that interspecies transmission has occurred on several occasions in the past. We investigated seropositive goats and sheep that were kept together in small mixed herds. Phylogenetic analysis of long proviral sequences in gag and pol, combined with epidemiologic information, demonstrated natural sheep-to-goat transmission of the recently identified SRLV subtype A4 in two instances and goat-to-sheep transmission of the same subtype in one instance. In a further mixed cluster, the direction of the interspecies transmission could not be determined. These findings present for the first time direct evidence that natural interspecies transmission of SRLV is ongoing in both directions. The findings are of relevance to virus eradication programs in both species