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In this note, we show that the asymptotic dimension of any building is finite and equal to the asymptotic dimension of an apartment in that building.
Skeletal polyhedra and polygonal complexes are finite or infinite periodic structures in 3-space with interesting geometric, combinatorial, and algebraic properties. These structures can be viewed as finite or infinite periodic graphs…
We prove an affine analog of Scharlau's reduction theorem for spherical buildings. To be a bit more precise let $X$ be a euclidean building with spherical building $\partial X$ at infinity. Then there exists a euclidean building $\bar X$…
An isomorphism of symplectically tame smooth pseudocomplex structures on the complex projective plane which is a homeomorphism and differentiable of full rank at two points is smooth.
A new construction of naturally reductive spaces is presented. This construction gives a large amount of new families of naturally reductive spaces. First the infinitesimal models of the new naturally reductive spaces are constructed. A…
In this paper we define spherical complexes as simplicial complexes with the property that every subcomplex obtained by a sequence of links and deletions either has trivial homology, or has the homology of a sphere. Examples of such…
The aim of this paper is to present the main constructions of the substructures of an almost groupoid and to discuss their basic properties. The definitions and properties concerning these new algebraic constructions extend to almost…
Similarly to the classic notion in $E^d$, a subset of a positive diameter below $\frac{\pi}{2}$ of a hemisphere of the sphere $S^d$ is called complete, provided adding any extra point increases its diameter. Complete sets are convex bodies…
We obtain several structure results for a class of spherical subgroups of connected reductive complex algebraic groups that extends the class of strongly solvable spherical subgroups. Based on these results, we construct certain…
Denote by $S^2$ the two-dimensional sphere. A spherical convex body on $S^2$ which does not properly contain a spherical convex body of the same spherical thickness is called a reduced body. We give three characterizations of reducedness of…
We prove that any convex flat subset in a complete Euclidean building is contained in an apartment of the maximal system of apartments.
We find the complete rational homology for the finite subset spaces of a $d$-dimensional sphere. We also determine the integral homology in top $d$ degrees and obtain a partial description of it in codimension $d$.
Every finite subunital of any generalized hermitian unital is itself a hermitian unital; the embedding is given by an embedding of quadratic field extensions. In particular, a generalized hermitian unital with a finite subunital is a…
Simple constructions are given for finite semifields that include as special cases both old semifields and recently constructed semifields.
The bottom complex of a finite polyhedal pointed rational cone is the lattice polytopal complex of the compact faces of the convex hull of nonzero lattice points in the cone. The algebra, associated to the bottom complex of a cone, defines…
A \emph{subcube partition} is a partition of the Boolean cube $\{0,1\}^n$ into subcubes. A subcube partition is irreducible if the only sub-partitions whose union is a subcube are singletons and the entire partition. A subcube partition is…
This note is the sequel of "Geometric structures as variational objects, I." It generalizes the main result and perspectives of that work to a class of geometric structures that includes integrable almost-complex structures.
Given a category, one may construct slices of it. That is, one builds a new category whose objects are the morphisms from the category with a fixed codomain and morphisms certain commutative triangles. If the category is a groupoid, so that…
The schematic finite spaces are those finite ringed spaces where a theory of quasi-coherent modules can be developed with minimal natural conditions. We give various characterizations of these spaces and their natural morphisms. We show…
We construct triangular hyperbolic polyhedra whose links are generalized 4-gons. The universal cover of those polyhedra are hyperbolic buildings, which appartments are hyperbolic planes tesselated by regular triangles with angles $\pi/4$.…