Abstract We prove a fixed point theorem for twin buildings of arbitrary rank. This theorem is then used to construct certain twin buildings whose existence was conjectured in [12]. As …
Abstract We prove a fixed point theorem for twin buildings of arbitrary rank. This theorem is then used to construct certain twin buildings whose existence was conjectured in [12]. As a consequence we obtain a classification of twin buildings whose rank 2 residues correspond to split algebraic groups over a field of cardinality at least 4. A similar result follows for twin buildings whose rank 2 residues are finite.
We investigate the geometry in a real Euclidean building X of type A2 of some simple configurations in the associated projective plane at infinity P, seen as ideal configurations in …
We investigate the geometry in a real Euclidean building X of type A2 of some simple configurations in the associated projective plane at infinity P, seen as ideal configurations in X, and relate it with the projective invariants (from the cross ratio on P). In particular we establish a geometric classification of generic triples of ideal chambers of X and relate it with the triple ratio of triples of flags.
In [5], Fox, Gromov, Lafforgue, Naor and Pach, in a response to a question of Gromov [7], constructed bounded degree geometric expanders, namely, simplical complexes having the affine overlapping property. …
In [5], Fox, Gromov, Lafforgue, Naor and Pach, in a response to a question of Gromov [7], constructed bounded degree geometric expanders, namely, simplical complexes having the affine overlapping property. Their explicit constructions are finite quotients of [Formula: see text]-buildings, for [Formula: see text], over local fields. In this paper, this result is extended to general high rank Bruhat–Tits buildings.
We investigate the possible ways in which a thick metasymplectic space Γ, that is, a Lie incidence geometry of type F4,1 (or F4,4), is embedded into the long root geometry …
We investigate the possible ways in which a thick metasymplectic space Γ, that is, a Lie incidence geometry of type F4,1 (or F4,4), is embedded into the long root geometry ∆ related to any building of type E7.We provide a complete classification (if Γ is not embedded in a singular subspace).As an application we prove the uniqueness of the inclusion of the long root geometry of type E6 in the one of type E7; it always arises as an equator geometry.We also use the latter concept to geometrically construct one of the embeddings turning up in our classification.As a side result we obtain that all triples of pairwise opposite elements of type 7 in a building of type E7 are projectively equivalent.
Completely reducible subcomplexes of spherical buildings was defined by J.P. Serre and are used in studying subgroups of reductive algebraic groups. We begin the study of completely reducible subcomplexes of …
Completely reducible subcomplexes of spherical buildings was defined by J.P. Serre and are used in studying subgroups of reductive algebraic groups. We begin the study of completely reducible subcomplexes of twin buildings and how they may be used to study subgroups of algebraic groups over a ring of Laurent polynomials and Kac-Moody groups by looking at the Euclidean twin building case.
Completely reducible subcomplexes of spherical buildings was defined by J.P. Serre and are used in studying subgroups of reductive algebraic groups. We begin the study of completely reducible subcomplexes of …
Completely reducible subcomplexes of spherical buildings was defined by J.P. Serre and are used in studying subgroups of reductive algebraic groups. We begin the study of completely reducible subcomplexes of twin buildings and how they may be used to study subgroups of algebraic groups over a ring of Laurent polynomials and Kac-Moody groups by looking at the Euclidean twin building case.
Let be a thick, spherical building equipped with its natural CAT(1) metric and let M be a proper, convex subset of .If M is open or if M is a …
Let be a thick, spherical building equipped with its natural CAT(1) metric and let M be a proper, convex subset of .If M is open or if M is a closed ball of radius =2, then ƒ, the maximal subcomplex supported by n M , is dim ƒ-spherical and non-contractible. 51E24; 11F75Connectivity properties of subcomplexes in spherical buildings play an important role in establishing finiteness properties of S -arithmetic groups.The complexes arise as relative links of filtrations of Euclidean buildings.The main result of this paper is the sphericity of open and closed hemisphere complexes in spherical buildings.
For a tame supercuspidal representation <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="pi"> <mml:semantics> <mml:mi>π<!-- π --></mml:mi> <mml:annotation encoding="application/x-tex">\pi</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of a connected reductive <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> …
For a tame supercuspidal representation <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="pi"> <mml:semantics> <mml:mi>π<!-- π --></mml:mi> <mml:annotation encoding="application/x-tex">\pi</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of a connected reductive <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-adic group <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, we establish two distinct and complementary sufficient conditions, formulated in terms of the geometry of the Bruhat–Tits building of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, for the irreducible components of its restriction to a maximal compact subgroup to occur in a representation of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> which is not inertially equivalent to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="pi"> <mml:semantics> <mml:mi>π<!-- π --></mml:mi> <mml:annotation encoding="application/x-tex">\pi</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. The consequence is a set of broadly applicable tools for addressing the branching rules of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="pi"> <mml:semantics> <mml:mi>π<!-- π --></mml:mi> <mml:annotation encoding="application/x-tex">\pi</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and the unicity of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-bracket upper G comma pi right-bracket Subscript upper G"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">[</mml:mo> <mml:mi>G</mml:mi> <mml:mo>,</mml:mo> <mml:mi>π<!-- π --></mml:mi> <mml:msub> <mml:mo stretchy="false">]</mml:mo> <mml:mi>G</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">[G,\pi ]_G</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-types.
The purpose of this paper is to provide an easier set of axioms for Affine $Λ$-Buildings by extending results of Anne Parreau on the equivalence of axioms for Euclidean buildings. …
The purpose of this paper is to provide an easier set of axioms for Affine $Λ$-Buildings by extending results of Anne Parreau on the equivalence of axioms for Euclidean buildings. In particular we give an easier set of axioms for an affine $Λ$-building, utilizing a notion of a strong exchange condition on apartments and sectors having a sector panel lying in an apartment.
An automorphism of a spherical building is called domestic if it maps no chamber to an opposite chamber. In this paper we classify domestic automorphisms of spherical buildings of classical …
An automorphism of a spherical building is called domestic if it maps no chamber to an opposite chamber. In this paper we classify domestic automorphisms of spherical buildings of classical type.
Let $k$ be a field, let $G$ be a reductive $k$-group and $V$ an affine $k$-variety on which $G$ acts. In this note we continue our study of the notion …
Let $k$ be a field, let $G$ be a reductive $k$-group and $V$ an affine $k$-variety on which $G$ acts. In this note we continue our study of the notion of cocharacter-closed $G(k)$-orbits in $V$. In earlier work we used a rationality condition on the point stabilizer of a $G$-orbit to prove Galois ascent/descent and Levi ascent/descent results concerning cocharacter-closure for the corresponding $G(k)$-orbit in $V$. In the present paper we employ building-theoretic techniques to derive analogous results.
Let $k$ be a field, let $G$ be a reductive $k$-group and $V$ an affine $k$-variety on which $G$ acts. In this note we continue our study of the notion …
Let $k$ be a field, let $G$ be a reductive $k$-group and $V$ an affine $k$-variety on which $G$ acts. In this note we continue our study of the notion of cocharacter-closed $G(k)$-orbits in $V$. In earlier work we used a rationality condition on the point stabilizer of a $G$-orbit to prove Galois ascent/descent and Levi ascent/descent results concerning cocharacter-closure for the corresponding $G(k)$-orbit in $V$. In the present paper we employ building-theoretic techniques to derive analogous results.
We now discuss a class of results related to universal constructions which we call the Hoffman-Newman-Radford (HNR) rigidity theorems. They come in different flavors which we explain one by one. …
We now discuss a class of results related to universal constructions which we call the Hoffman-Newman-Radford (HNR) rigidity theorems. They come in different flavors which we explain one by one. In each case, the result provides explicit inverse isomorphisms between two universally constructed bimonoids. We call these the Hoffman-Newman-Radford (HNR) isomorphisms. For a cocommutative comonoid, the free bimonoid on that comonoid is isomorphic to the free bimonoid on the same comonoid but with the trivial coproduct. The product is concatenation in both, but the coproducts differ, it is dequasishuffle in the former and deshuffle in the latter. An explicit isomorphism can be constructed in either direction, one direction involves a noncommutative zeta function, while the other direction involves a noncommutative Möbius function.These are the HNR isomorphisms. There is a dual result starting with a commutative monoid.In this case, the coproduct is deconcatenation in both, but the products differ, it is quasishuffle in the former and shuffle in the latter. Interestingly, these ideas can be used to prove that noncommutative zeta functions and noncommutative Möbius functions are inverse to each other in the lune-incidence algebra. There is a commutative analogue of the above results in which the universally constructed bimonoids are bicommutative. Now the HNR isomorphisms are constructed using the zeta function and Möbius function of the poset of flats. As an application, we explain how they can be used to diagonalize the mixed distributive law for bicommutative bimonoids. There is also a q-analogue, for q not a root of unity. In this case, the HNR isomorphisms involve the two-sided q-zeta and q-Möbius functions. As an application, we explain how they can be used to study the nondegeneracy of the mixed distributive law for q-bimonoids.
The $1/3$-$2/3$ Conjecture, originally formulated in 1968, is one of the best-known open problems in the theory of posets, stating that the balance constant (a quantity determined by the linear …
The $1/3$-$2/3$ Conjecture, originally formulated in 1968, is one of the best-known open problems in the theory of posets, stating that the balance constant (a quantity determined by the linear extensions) of any non-total order is at least $1/3$. By reinterpreting balance constants of posets in terms of convex subsets of the symmetric group, we extend the study of balance constants to convex subsets $C$ of any Coxeter group. Remarkably, we conjecture that the lower bound of $1/3$ still applies in any finite Weyl group, with new and interesting equality cases appearing. We generalize several of the main results towards the $1/3$-$2/3$ Conjecture to this new setting: we prove our conjecture when $C$ is a weak order interval below a fully commutative element in any acyclic Coxeter group (an generalization of the case of width-two posets), we give a uniform lower bound for balance constants in all finite Weyl groups using a new generalization of order polytopes to this context, and we introduce generalized semiorders for which we resolve the conjecture. We hope this new perspective may shed light on the proper level of generality in which to consider the $1/3$-$2/3$ Conjecture, and therefore on which methods are likely to be successful in resolving it.
In this paper we consider various problems involving the action of a reductive group G on an affine variety V . We prove some general rationality results about the G-orbits …
In this paper we consider various problems involving the action of a reductive group G on an affine variety V . We prove some general rationality results about the G-orbits in V . In addition, we extend fundamental results of Kempf and Hesselink regarding optimal destabilizing parabolic subgroups of G for such general G-actions. We apply our general rationality results to answer a question of Serre concerning how his notion of G-complete reducibility behaves under separable field extensions. Applications of our new optimality results also include a construction which allows us to associate an optimal destabilizing parabolic subgroup of G to any subgroup of G. Finally, we use these new optimality techniques to provide an answer to Tits’ Centre Conjecture in a special case.
In a celebrated paper ``Can one hear the shape of a drum?'' M. Kac [Am. Math. Monthly 73, 1 (1966)] asked his famous question about the existence of nonisometric billiards …
In a celebrated paper ``Can one hear the shape of a drum?'' M. Kac [Am. Math. Monthly 73, 1 (1966)] asked his famous question about the existence of nonisometric billiards having the same spectrum of the Laplacian. This question was eventually answered positively in 1992 by the construction of noncongruent planar isospectral pairs. This review highlights mathematical and physical aspects of isospectrality.
In this paper, we study classes of discrete convex functions: submodular functions on modular semilattices and L-convex functions on oriented modular graphs. They were introduced by the author in complexity …
In this paper, we study classes of discrete convex functions: submodular functions on modular semilattices and L-convex functions on oriented modular graphs. They were introduced by the author in complexity classification of minimum 0-extension problems. We clarify the relationship to other discrete convex functions, such as $k$-submodular functions, skew-bisubmodular functions, L$^\natural$-convex functions, tree submodular functions, and UJ-convex functions. We show that they are actually viewed as submodular/L-convex functions in our sense. We also prove a sharp iteration bound of the steepest descent algorithm for minimizing our L-convex functions. The underlying structures, modular semilattices and oriented modular graphs, have rich connections to projective and polar spaces, Euclidean building, and metric spaces of global nonpositive curvature (CAT(0) spaces). By utilizing these connections, we formulate an analogue of the Lov\'asz extension, introduce well-behaved subclasses of submodular/L-convex functions, and show that these classes can be characterized by the convexity of the Lov\'asz extension. We demonstrate applications of our theory to combinatorial optimization problems that include multicommodity flow, multiway cut, and related labeling problems: these problems have been outside the scope of discrete convex analysis so far.
We study $2$-dimensional Artin groups of hyperbolic type from the viewpoint of measure equivalence, and establish rigidity theorems. We first prove that they are boundary amenable. So is every group …
We study $2$-dimensional Artin groups of hyperbolic type from the viewpoint of measure equivalence, and establish rigidity theorems. We first prove that they are boundary amenable. So is every group acting discretely by simplicial isometries on a connected piecewise hyperbolic $\mathrm{CAT}(-1)$ simplicial complex with countably many simplices in finitely many isometry types, assuming that vertex stabilizers are boundary amenable. Consequently, they satisfy the Novikov conjecture. We then show that measure equivalent $2$-dimensional Artin groups of hyperbolic type have isomorphic fixed set graphs -- an analogue of the curve graph, introduced by Crisp. This yields classification results. We obtain strong rigidity theorems. Let $G=G_\Gamma$ be a $2$-dimensional Artin group of hyperbolic type, with $\mathrm{Out}(G)$ finite. When the automorphism groups of the fixed set graph and of the Cayley complex $\mathfrak{C}$ coincide, every countable group $H$ which is measure equivalent to $G$, is commensurable to a lattice in $\mathrm{Aut}(\mathfrak{C})$. This happens whenever $\Gamma$ is triangle-free with all labels at least $3$ -- unless $G$ is commensurable to the direct sum of $\mathbb{Z}$ and a free group. When $\Gamma$ satisfies an additional star-rigidity condition, then $\mathrm{Aut}(\mathfrak{C})$ is countable, and $H$ is almost isomorphic to $G$. This has applications to orbit equivalence rigidity, and rigidity results for von Neumann algebras associated to ergodic actions of Artin groups. We also derive a rigidity statement regarding possible lattice envelopes of certain Artin groups, and a cocycle superrigidity theorem from higher-rank lattices to $2$-dimensional Artin groups of hyperbolic type.
A notion of âabstract reflectionâ is introduced and used to characterize the Coxeter groups. Several known results on Coxeter groups are obtained as easy corollaries.
A notion of âabstract reflectionâ is introduced and used to characterize the Coxeter groups. Several known results on Coxeter groups are obtained as easy corollaries.
Let Y be an abelian variety of dimensiongover an algebraically closed fieldkof characteristicp >0. To Y we can associate its p-kernel Y[p], which is a finite commutative k-group scheme of …
Let Y be an abelian variety of dimensiongover an algebraically closed fieldkof characteristicp >0. To Y we can associate its p-kernel Y[p], which is a finite commutative k-group scheme of rank p2 gg. In the unpublished manuscript [8], Kraft showed that, fixinggthere are only finitely many such group schemes, up to isomorphism. (As we shall discuss later, Kraft also gave a very nice description of all possible types.) About 20 years later, this result was re-obtained, independently, by Oort. Together with Ekedahl he used it to define and study a stratification of the moduli space akof principally polarized abelian varieties overk.The strata correspond to the pairs (Y, such that the p-kernel is of a fixed isomorphism type. Their results can be found in [11] and [12]; see also related work by van der Geer in [, [17].
We study stability properties of the Haagerup Property and of coarse embeddability in a Hilbert space, under certain semidirect products. In particular, we prove that they are stable under taking …
We study stability properties of the Haagerup Property and of coarse embeddability in a Hilbert space, under certain semidirect products. In particular, we prove that they are stable under taking standard wreath products. Our construction also provides a characterization of subsets with relative Property T in a standard wreath product.
We call a non-discrete Euclidean building a {\it Bruhat-Tits space} if its automorphism group contains a subgroup that induces the subgroup generated by all the root groups of a root …
We call a non-discrete Euclidean building a {\it 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 $\hbox{\sf B}_2$, $\hbox{\sf F}_4$ or $\hbox{\sf G}_2$ associated with a Ree or Suzuki group endowed with the usual root datum. (In the $\hbox{\sf B}_2$ and $\hbox{\sf G}_2$ cases, this fixed point set is a building of rank one; in the $\hbox{\sf F}_4$ 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.
A. Reid showed that if $Γ_1$ and $Γ_2$ are arithmetic lattices in $G = \operatorname{PGL}_2(\mathbb R)$ or in $\operatorname{PGL}_2(\mathbb C)$ which give rise to isospectral manifolds, then $Γ_1$ and $Γ_2$ …
A. Reid showed that if $Γ_1$ and $Γ_2$ are arithmetic lattices in $G = \operatorname{PGL}_2(\mathbb R)$ or in $\operatorname{PGL}_2(\mathbb C)$ which give rise to isospectral manifolds, then $Γ_1$ and $Γ_2$ are commensurable (after conjugation). We show that for $d \geq 3$ and ${\mathcal S} = \operatorname{PGL}_d(\mathbb R) / \operatorname{PGO}_d(\mathbb R)$, or ${\mathcal S} = \operatorname{PGL}_d(\mathbb C) / \operatorname{PU}_d(\mathbb C)$, the situation is quite different: there are arbitrarily large finite families of isospectral non-commensurable compact manifolds covered by $\mathcal S$. The constructions are based on the arithmetic groups obtained from division algebras with the same ramification points but different invariants.
We prove that even Coxeter groups, whose Coxeter diagrams contain no $(4,4,2)$ triangles, are conjugacy separable. In particular, this applies to all right-angled Coxeter groups or word hyperbolic even Coxeter …
We prove that even Coxeter groups, whose Coxeter diagrams contain no $(4,4,2)$ triangles, are conjugacy separable. In particular, this applies to all right-angled Coxeter groups or word hyperbolic even Coxeter groups. For an arbitrary Coxeter group $W$, we also study the relationship between Coxeter generating sets that give rise to the same collection of parabolic subgroups. As an application, we show that if an automorphism of $W$ preserves the conjugacy class of every sufficiently short element then it is inner. We then derive consequences for the outer automorphism groups of Coxeter groups.
Let <italic>G</italic> be a finite (<italic>B, N</italic>)-pair whose Coxeter system is of type <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E 6"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>E</mml:mi> <mml:mn>6</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{E_6}</mml:annotation> …
Let <italic>G</italic> be a finite (<italic>B, N</italic>)-pair whose Coxeter system is of type <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E 6"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>E</mml:mi> <mml:mn>6</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{E_6}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> or <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E 7"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>E</mml:mi> <mml:mn>7</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{E_7}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="1 Subscript upper B Superscript upper G"> <mml:semantics> <mml:msubsup> <mml:mn>1</mml:mn> <mml:mi>B</mml:mi> <mml:mi>G</mml:mi> </mml:msubsup> <mml:annotation encoding="application/x-tex">1_B^G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be the permutation character of the action of <italic>G</italic> on the left cosets of the Borel subgroup <italic>B</italic> in <italic>G</italic>. In this paper we give the character degrees of the irreducible constituents of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="1 Subscript upper B Superscript upper G"> <mml:semantics> <mml:msubsup> <mml:mn>1</mml:mn> <mml:mi>B</mml:mi> <mml:mi>G</mml:mi> </mml:msubsup> <mml:annotation encoding="application/x-tex">1_B^G</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.
One says that Veldkamp lines exist for a point-line geometry if, for any three distinct (geometric) hyperplanes A, B and C (i) is not properly contained in B and (ii) …
One says that Veldkamp lines exist for a point-line geometry if, for any three distinct (geometric) hyperplanes A, B and C (i) is not properly contained in B and (ii) A\B C implies C or A B = A\C. Under this condition, the set V of all hyperplanes of acquires the structure of a linear space { the Veldkamp space { with intersections of distinct hyperplanes playing the role of lines. It is shown here that an interesting class of strong parapolar spaces (which includes both the half-spin geometries and the Grassmannians) possess Veldkamp lines. Combined with other results on hyperplanes and embeddings, this implies that for most of these parapolar spaces, the corresponding Veldkamp spaces are projective spaces. The arguments incorporate a model of partial matroids based on intersections of sets.
In this short note, completing a sequence of studies, we consider the $k$-Grassmannians of a number of polar geometries of finite rank $n$. We classify those subspaces that are isomorphic …
In this short note, completing a sequence of studies, we consider the $k$-Grassmannians of a number of polar geometries of finite rank $n$. We classify those subspaces that are isomorphic to the $j$-Grassmannian of a projective $m$-space. In almost all cases, these are parabolic, that is, they are the residues of a flag of the polar geometry. Exceptions only occur when the subspace is isomorphic to the Grassmannian of $2$-spaces in a projective $m$-space and we describe these in some detail. This Witt-type result implies that automorphisms of the Grassmannian are almost always induced by automorphisms of the underlying polar space.
The paper studies various relationships between locally toroidal regular 4-polytopes of types {6, 3,p} and {3, 6, 3}. These relationships are based on corresponding relationships between the regular honeycombs with …
The paper studies various relationships between locally toroidal regular 4-polytopes of types {6, 3,p} and {3, 6, 3}. These relationships are based on corresponding relationships between the regular honeycombs with the same Schläfli-symbol in hyperbolic 3-space. Also the paper discusses regular tessellations (secretions of rank 3) which are locally inscribed into regular 4-polytopes. In particular, this leads to local criteria for the finiteness of the polytopes.
We study final group topologies and their relations to compactness properties. In particular, we are interested in situations where a colimit or direct limit is locally compact, a k_\omega-space, or …
We study final group topologies and their relations to compactness properties. In particular, we are interested in situations where a colimit or direct limit is locally compact, a k_\omega-space, or locally k_\omega. As a first application, we show that unitary forms of complex Kac-Moody groups can be described as the colimit of an amalgam of subgroups (in the category of Hausdorff topological groups, and the category of k_\omega-groups). Our second application concerns Pontryagin duality theory for the classes of almost metrizable topological abelian groups, resp., locally k_\omega topological abelian groups, which are dual to each other. In particular, we explore the relations between countable projective limits of almost metrizable abelian groups and countable direct limits of locally k_\omega abelian groups.