A sixteen-relator presentation of an infinite hyperbolic Kazhdan group

We provide an explicit presentation of an infinite hyperbolic Kazhdan group with $4$ generators and $16$ relators of length at most $73$. That group acts properly and cocompactly on a hyperbolic triangle building of type $(3,4,4)$. We also point out a variation of the construction that yields examples of lattices in $\tilde A_2$-buildings admitting non-Desarguesian residues of arbitrary prime power order.Comment: 9 pages, 1 figur

Automorphism groups of right-angled buildings: simplicity and local splittings

We show that the group of type-preserving automorphisms of any irreducible semi-regular thick right-angled building is abstractly simple. When the building is locally finite, this gives a large family of compactly generated (abstractly) simple locally compact groups. Specializing to appropriate cases, we obtain examples of such simple groups that are locally indecomposable, but have locally normal subgroups decomposing non-trivially as direct products.Comment: 26 pages. Several points were clarified and a few lemmas were added, in accordance with the referee's repor

Amenable groups and Hadamard spaces with a totally disconnected isometry group

Let $X$ be a locally compact Hadamard space and $G$ be a totally disconnected group acting continuously, properly and cocompactly on $X$. We show that a closed subgroup of $G$ is amenable if and only if it is (topologically locally finite)-by-(virtually abelian). We are led to consider a set \bdfine X which is a refinement of the visual boundary \bd X. For each x \in \bdfine X, the stabilizer $G_x$ is amenable.Comment: 15 page

A uniform bound on the nilpotency degree of certain subalgebras of Kac-Moody algebras

Let $\mathfrak{g}$ be a Kac-Moody algebra and $\mathfrak{b}_1, \mathfrak{b}_2$ be Borel subalgebras of opposite signs. The intersection $\mathfrak{b} = \mathfrak{b}_1 \cap \mathfrak{b}_2$ is a finite-dimensional solvable subalgebra of $\mathfrak{g}$. We show that the nilpotency degree of $[\mathfrak{b}, \mathfrak{b}]$ is bounded from above by a constant depending only on $\mathfrak{g}$. This confirms a conjecture of Y. Billig and A. Pianzola \cite{BilligPia95}

Geometry over composition algebras : projective geometry

The purpose of this article is to introduce projective geometry over composition algebras : the equivalent of projective spaces and Grassmannians over them are defined. It will follow from this definition that the projective spaces are in correspondance with Jordan algebras and that the points of a projective space correspond to rank one matrices in the Jordan algebra. A second part thus studies properties of rank one matrices. Finally, subvarieties of projective spaces are discussed.Comment: 24 page

On Mukai flops for Scorza varieties

I give three descriptions of the Mukai flop of type $E\_{6,I}$, one in terms of Jordan algebras, one in terms of projective geometry over the octonions, and one in terms of O-blow-ups. Each description shows that it is very similar to certain flops of type $A$. The Mukai flop of type $E\_{6,II}$ is also described.Comment: 35

A 3D discrete model of the diaphragm and human trunk

In this paper, a 3D discrete model is presented to model the movements of the trunk during breathing. In this model, objects are represented by physical particles on their contours. A simple notion of force generated by a linear actuator allows the model to create forces on each particle by way of a geometrical attractor. Tissue elasticity and contractility are modeled by local shape memory and muscular fibers attractors. A specific dynamic MRI study was used to build a simple trunk model comprised of by three compartments: lungs, diaphragm and abdomen. This model was registered on the real geometry. Simulation results were compared qualitatively as well as quantitatively to the experimental data, in terms of volume and geometry. A good correlation was obtained between the model and the real data. Thanks to this model, pathology such as hemidiaphragm paralysis can also be simulated.Comment: published in: "Lung Modelling", France (2006