Large Almost Simple Groups Acting on 16-Dimensional Compact Projective Planes

 This paper deals with the so-called Salzmann program aiming to classify special geometries according to their automorphism groups. Here, topological connected compact projective planes are considered. If finite-dimensional, such planes are of dimension 2, 4, 8, or 16. The classical example of a 16-dimensional, compact projective plane is the projective plane over the octonions with 78-dimensional automorphism group E₆(−26). A 16-dimensional, compact projective plane ? admitting an automorphism group of dimension 41 or more is clasical, [23] 87.5 and 87.7. For the special case of a semisimple group Δ acting on ? the same result can be obtained if dim , see [22]. Our aim is to lower this bound. We show: if Δ is semisimple and dim , then ? is either classical or a Moufang-Hughes plane or Δ is isomorphic to Spin₉ (ℝ, r), r∈{0, 1}. The proof consists of two parts. In [16] it has been shown that Δ is in fact almost simple or isomorphic to SL₃?ċSpin₃ℝ. In the underlying paper we can therefore restrict our considerations to the case that Δ is almost simple, and the corresponding planes are classified.     

Large semisimple groups on 16-dimensional compact projective planes are almost simple

The paper deals with the so-called Salzmann program aiming to classify special geometries according to their automorphism groups. Here, topological connected compact projective planes are considered. If finite-dimensional, such planes are of dimension 2, 4, 8, or 16. The classical example of a 16-dimensional, compact projective plane is the projective plane over the octonions with 78-dimensional automorphism group E₆(?26). A 16-dimensional, compact projective plane P admitting an automorphism group of dimension 41 or more is classical, [18] 87.5 and 87.7. For the special case of a semisimple group Δ acting on P the same result can be obtained if dim δ ≧ 37, see [16]. Our aim is to lower this bound. We show: if Δ is semisimple and dim δ ≧ 29, then P is either classical or a Moufang-Hughes plane or Δ is isomorphic to Spin₉ (?, r), r ∈ {0, 1 }. The underlying paper contains the first part of the proof showing that Δ is in fact almost simple.     

Compact projective planes with homogeneous ovals

Closed ovals exist only in 2-or 4-dimensional compact projective planes. We show that a plane of dimension 2 or 4 contains ahomogeneous closed oval iff the automorphism group contains SO₂ or SO₃, respectively. In 4-dimensional planes, existence of a homogeneous oval and existence of a homogeneous Baer subplane are equivalent. We determine the possible full automorphism groups of planes containing homogeneous ovals except for two possibilities in dimension 4 whose existence remains uncertain.     

Smooth Hughes planes are classical

We prove that the only compact projective Hughes planes which are smooth projective planes are the classical planes over the complex numbers \Bbb C \Bbb C , the quaternions \Bbb H \Bbb H , and the Caley numbers \Bbb O \Bbb O . As a by-product this shows that an 8-dimensional smooth projective plane which admits a collineation group of dimension d 3 17d \geq 17 is isomorphic to the quaternion projective plane P ₂\Bbb H {\cal P _2\Bbb H }. For topological compact projective planes this is true if d 3 19d \geq 19, and this bound is sharp.     

Realizability of the torus and the projective plane in ℝ4

We show that every triangulation of the projective plane or the torus is isomorphic to a subcomplex of the boundary complex of a simplicial 5-dimensional convex polytope and thus linearly embeddable in ℝ⁴.     

16-Dimensional Smooth Projective Planes with Large Collineation Groups

Smooth projective planes are projective planes defined on smooth manifolds (i.e. the set of points and the set of lines are smooth manifolds) such that the geometric operations of join and intersection are smooth. A systematic study of such planes and of their collineation groups can be found in previous works of the author. We prove in this paper that a 16-dimensional smooth projective plane which admits a collineation group of dimension d 39 is isomorphic to the octonion projective plane P₂ O. For topological compact projective planes this is true if d 41. Note that there are nonclassical topological planes with a collineation group of dimension 40.     

Shear planes

A shear plane is a 2n-dimensional stable plane admitting a quasi-perspective collineation group which is a vector group of the same dimension 2n and fixes no point. We show that all of these planes can be derived from a special kind of partial spreads by a construction analogous to the construction of (punctured) dual translation planes from compact spreads. Finally we give a criterion (and examples) for shear planes which are not isomorphic to an open subplane of a topological projective plane.     

A minimal triangulation of complex projective plane admitting a chess colouring of four-dimensional simplices

We construct and study a new 15-vertex triangulation X of the complex projective plane ℂP². The automorphism group of X is isomorphic to S ₄ × S ₃. We prove that the triangulation X is the minimal (with respect to the number of vertices) triangulation of ℂP² admitting a chess colouring of four-dimensional simplices. We provide explicit parametrizations for the simplices of X and show that the automorphism group of X can be realized as a group of isometries of the Fubini-Study metric. We find a 33-vertex subdivision $ \bar X $ \bar X of the triangulation X such that the classical moment mapping μ: ℂP² → Δ² is a simplicial mapping of the triangulation $ \bar X $ \bar X onto the barycentric subdivision of the triangle Δ². We study the relationship of the triangulation X with complex crystallographic groups.     

A Remark on a Theorem by Kodama and Shimizu

We prove a characterization theorem for the unit polydisc Δ ⁿ ⊂ℂ ⁿ in the spirit of a recent result due to Kodama and Shimizu. We show that if M is a connected n-dimensional complex manifold such that (i) the group Aut (M) of holomorphic automorphisms of M acts on M with compact isotropy subgroups, and (ii) Aut (M) and Aut (Δ ⁿ ) are isomorphic as topological groups equipped with the compact-open topology, then M is holomorphically equivalent to Δ ⁿ .      

Sixteen-Dimensional Locally Compact Translation Planes with Large Automorphism Groups having no Fixed Points

The 16-dimensional compact projective planes whose automorphism group contains a closed connected subgroup fixing a line, but no point and having dimension at least 35 are determined. It is shown that these planes all belong to three families of planes determined by H. Löwe and the author, and hence are explicitly known. A major stepping stone to this goal is a result by H. Salzmann according to which every such plane is a translation plane.     

Complex<Emphasis Type="BoldItalic">n</Emphasis>-dimensional manifolds with a real<Emphasis Type="BoldItalic">n</Emphasis><Superscript>2</Superscript>-dimensional automorphism group

The main theorem of this article is a characterization of non compact simply connected complete Kobayashi hyperbolic complex manifold of dimension n≽ 2 with real n ²-dimensional holomorphic automorphism group. Together with the earlier work [11, 12] and [13] of Isaev and Krantz, this yields a complete classification of the simply-connected, complete Kobayashi hyperbolic manifolds with dim_ℝ Aut (M) ≽ (dim_ℂ M)².     

Four-dimensional compact projective planes with a nonsolvable automorphism group

We contribute to the enumeration of all four-dimensional compact projective planes with an at least seven-dimensional automorphism group (cf. Betten [8]) by treating the nonsolvable case. Moreover, we find that the only possible six-dimensional nonsolvable automorphism group is ² · GL ₊ ² .Dedicated to Professor H. Salzmann on his 60th birthday     

Large automorphism groups of 8-dimensional projective planes are Lie groups

We prove the following theorem: LetP be an 8-dimensional compact topological projective plane. If the connected component of its automorphism group has dimension at least 12, then is a Lie group.     

Achtdimensionale lokalkompakte Translationsebenen mit großen kompakten Kollineationsgruppen

This paper is part of a program aiming at the classification of all higher-dimensional locally compact translation planes whose collineation groups have large dimension. In the present paper we determine all eight-dimensional locally compact translation planes which admit acompact collineation group of dimension at least 5 acting almost effectively on the translation axis. In fact, is isomorphic either to Spin₄ or toSO ₄(). The case Spin₄() has already been treated elsewhere ([6]). Here, the planes with SO ₄() are explicitly determined and studied in detail.     

Absolute continuity of the spectrum of the periodic Schrödinger operator in a layer and in a smooth cylinder

The Schr?dinger operator H = −Δ + V is considered in a layer or in a d-dimensional cylinder. The potential V is assumed to be periodic with respect to a lattice. The absolute continuity of H is established, provided that V ∈ L _p,loc, where p is a real number greater than d/2 in the case of a layer and p > max(d/2, d − 2) for a cylinder. Bibliography: 14 titles.     

A characterization of the higher dimensional groups associated with continuous wavelets

A subgroup D of GL (n, ℝ) is said to be admissible if the semidirect product of D and ℝ ⁿ , considered as a subgroup of the affine group on ℝ ⁿ , admits wavelets ψ ∈ L²(ℝ ⁿ ) satisfying a generalization of the Calderón reproducing, formula. This article provides a nearly complete characterization of the admissible subgroups D. More precisely, if D is admissible, then the stability subgroup D_x for the transpose action of D on ℝ ⁿ must be compact for a. e. x. ∈ ℝ ⁿ ; moreover, if Δ is the modular function of D, there must exist an a ∈ D such that |det a| ≠ Δ(a). Conversely, if the last condition holds and for a. e. x ∈ ℝ ⁿ there exists an ε > 0 for which the ε-stabilizer D _x ^ε is compact, then D is admissible. Numerous examples are given of both admissible and non-admissible groups.     

Die Oktavenebene als Translationsebene mit großer Kollineationsgruppe

It is shown that the affine plane over the Cayley numbers is the only 16-dimensional locally compact topological translation plane having a collineation group of dimension at least 41. This (hitherto unpublished) result is one of the ingredients of H. Salzmann's characterizations of the Cayley plane among general compact projective planes by the size of its collineation group.The proof involves various case studies of the possibilities for the structure and size of collineation groups of 16-dimensional locally compact translation planes. At the same time, these case studies are important steps for a classification program aiming at the explicit determination of all such translation planes having a collineation group of dimension at least 38.     

4-Dimensional Elation Laguerre Planes Admitting Non-Solvable Automorphism Groups

 This paper concerns 4-dimensional (topological locally compact connected) elation Laguerre planes that admit non-solvable automorphism groups. It is shown that such a plane is either semi-classical or a single plane admitting the group SL(2, ). Various characterizations of this single Laguerre plane are obtained. Received October 17 2000; in revised form April 23 2001 Published online August 5, 2002     

Eine Klasse von achtdimensionalen lokalkompakten Translationsebenen mit großen Scherungsgruppen

This paper is one of the final steps in a classification program to determine all eight-dimensional, locally compact translation planes having large collineation groups. Here, we describe all such planes whose collineation group contains a semidirect product ·N, whereN is an at least 3-dimensional normal subgroup consisting of shears with fixed axis, and is isomorphic to SO₃ ().     

Almost sure asymptotics for the continuous parabolic Anderson model

We consider the parabolic Anderson problem ∂ _t u = κΔu + ξ(x)u on ℝ₊×ℝ ^d with initial condition u(0,x) = 1. Here κ > 0 is a diffusion constant and ξ is a random homogeneous potential. We concentrate on the two important cases of a Gaussian potential and a shot noise Poisson potential. Under some mild regularity assumptions, we derive the second-order term of the almost sure asymptotics of u(t, 0) as t→∞. Received: 26 July 1999 / Revised version: 6 April 2000 / Published online: 22 November 2000