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Linus Kramer

Publications and source records attributed to Linus Kramer.

36 records · Page 2Linked to original sources
arXiv2011

Compact Totally Disconnected Moufang Buildings

Let $Δ$ be a spherical building each of whose irreducible components is infinite, has rank at least 2 and satisfies the Moufang condition. We show that $Δ$ can be given the structure of a topological building that is compact and totally disconnected precisely when $Δ$ is the building at infinity of a locally finite affine building.

math.GR↗
arXiv2011

The topology of a semisimple Lie group is essentially unique

We study locally compact group topologies on semisimple Lie groups. We show that the Lie group topology on such a group $S$ is very rigid: every 'abstract' isomorphism between $S$ and a locally compact and $σ$-compact group $Γ$ is automatically a homeomorphism, provided that $S$ is absolutely simple. If $S$ is complex, then non-continuous field automorphisms of the complex numbers have to be considered, but that is all.

math.GR↗
arXiv2010

On the local structure and the homology of CAT$(κ)$ spaces and euclidean buildings

We prove that every open subset of a euclidean building is a finite dimensional absolute neighborhood retract. This implies in particular that such a set has the homotopy type of a finite dimensional simplicial complex. We also include a proof for the rigidity of homeomorphisms of euclidean buildings. A key step in our approach to this result is the following: the space of directions $Σ_oX$ of a CAT$(κ)$ space $X$ is homotopy quivalent to a small punctured disk $B_\eps(X,o)\setminus o$. The second ingredient is the local homology sheaf of $X$. Along the way, we prove some results about the local structure of CAT$(κ)$-spaces which may be of independent interest.

math.MG↗
arXiv2009

A Maslov cocycle for unitary groups

We introduce a 2-cocycle for symplectic and skew-hermitian hyperbolic groups over arbitrary fields and skew fields, with values in the Witt group of hermitian forms. This cocycle has good functorial properties: it is natural under extension of scalars and stable, so it can be viewed as a universal 2-dimensional characteristic class for these groups. Over R and C, it coincides with the first Chern class.

math.KT↗
arXiv2009

Non-discrete Euclidean Buildings for the Ree and Suzuki groups

We call a non-discrete Euclidean building a Bruhat-Tits space if its automorphism group contains a subgroup that induces the subgroup generated by all the root groups of a root datum of the building at infinity. This is the class of non-discrete Euclidean buildings introduced and studied by Bruhat and Tits. We give the complete classification of Bruhat-Tits spaces whose building at infinity is the fixed point set of a polarity of an ambient building of type B_2, F_4 or G_2 associated with a Ree or Suzuki group endowed with the usual root datum. (In the B_2 and G_2 cases, this fixed point set is a building of rank one; in the F_4 case, it is a generalized octagon whose Weyl group is not crystallographic.) We also show that each of these Bruhat-Tits spaces has a natural embedding in the unique Bruhat-Tits space whose building at infinity is the corresponding ambient building.

math.GR↗
arXiv2008

The real quadrangle of type E6

We give a geometric interpretation of the building associated to the real Lie group E_6(-14) in terms of its 54-dimensional module.

math.GR↗
arXiv2004

Asymptotic cones of finitely presented groups

Let G be a connected semisimple Lie group with at least one absolutely simple factor S such that R-rank(S) is at least 2, and let $Γ$ be a uniform lattice in G. (a) If $CH$ holds, then $Γ$ has a unique asymptotic cone up to homeomorphism. (b) If $CH$ fails, then $Γ$ has $2^{2^ω}$ asymptotic cones up to homeomorphism.

math.GT↗
arXiv2004

Asymptotic cones and ultrapowers of Lie groups

Asymptotic cones of metric spaces were first invented by Gromov. They are metric spaces which capture the 'large-scale structure' of the underlying metric space. Later, van den Dries and Wilkie gave a more general construction of asymptotic cones using ultrapowers. Certain facts about asymptotic cones, like the completeness of the metric space, now follow rather easily from saturation properties of ultrapowers, and in this survey, we want to present two applications of the van den Dries-Wilkie approach. Using ultrapowers we obtain an explicit description of the asymptotic cone of a semisimple Lie group. From this description, using semi-algebraic groups and non-standard methods, we can give a short proof of the Margulis Conjecture. In a second application, we use set theory to answer a question of Gromov.

math.GT↗
arXiv2003

Projective planes and their look-alikes

We classify all closed 1-connected manifolds $M$ which look like projective planes, i.e. with integral homology $H_*(M)=Z^3$. Furthermore, we give an explicit construction of these manifolds as Thom spaces of open disk bundles.

math.GT↗
arXiv2003

Two-transitive Lie groups

Using a characterization of parabolics in reductive Lie groups due to Furstenberg, elementary properties of buildings, and some algebraic topology, we give a new proof of Tits' classification of 2-transitive Lie groups.

math.DG↗
arXiv2002

Affine $Λ$-buildings, ultrapowers of Lie groups and Riemannian symmetric spaces: an algebraic proof of the Margulis conjecture

In this paper, we give a general group-theoretic construction of affine $\RR$-buildings, and more generally, of affine $Λ$-buildings, associated to semisimple Lie groups over nonarchimedean real closed fields. The construction of Kleiner-Leeb using the asymptotic cone of a Riemannian symmetric space appears as a special case. The explicit knowledge of the building arising here as the asymptotic cone simplifies the proof of the Margulis conjecture due to Kleiner-Leeb.

math.DG↗
arXiv2001

Loop groups and twin buildings

We describe some buildings related to complex Kac-Moody groups. First we describe the spherical building of SLn(C) (i.e. the projective geometry PG(Cn)) and its Veronese representation. Next we recall the construction of the affine building associated to a discrete valuation on the rational function field $C(z)$. Then we describe the same building in terms of complex Laurent polynomials, and introduce the Veronese representation, which is an equivariant embedding of the building into an affine Kac-Moody algebra. Next, we introduce topological twin buildings. These buildings can be used for a proof - which is a variant of the proof by Quillen and Mitchell - of Bott periodicity which uses only topological geometry. At the end we indicate very briefly that the whole process works also for affine real almost split Kac-Moody groups.

math.GT↗
arXiv2001

Homogeneous spaces, Tits buildings, and isoparametric hypersurfaces

We classify compact 2-connected homogeneous spaces with the same rational cohomology as a product of spheres. This classification relies on spectral sequences, homotopy theory, and representation theory. We then apply this classification to two geometric problems. The first problem is the classification of all isoparametric hypersurfaces which admit a transitive isometry group on at least one focal manifold. This generalizes the classification of homogeneous isoparametric hypersurfaces by Hsiang and Lawson and gives a new, independent proof of their result. Secondly, we classify certain compact highly connected Tits buildings which admit a vertex transitive automorphism group. Such buildings arise as compactifications of symmetric spaces as well as from isoparametric submanifolds. This extends the recent classification of all compact connected Tits buildings which admit a chamber transitive automorphism group by Grundhofer, Knarr, and the author.

math.GT↗
arXiv2001

Compact Polygons

We develop the basic topological properties of compact polygons, i.e. of compact topological Tits buildings of rank two. It is proved that the Coxeter diagram of such a building is always crystallographic, that is, compact connected n-gons exist only for n=3,4,6. We classify compact polygons which admit a transitive group action, showing that such a polygon is Moufang and thus related to a real Lie group of rank 2.

math.DG↗