675 research outputs found
Vertex opposition in spherical buildings
We study to which extent all pairs of opposite vertices of self-opposite type determine a given building. We provide complete answers in the case of buildings related to projective spaces, to polar spaces and the exceptional buildings, but for the latter we restrict to the vertices whose Grassmannian defines a parapolar space of point diameter 3. Some results about non-self opposite types for buildings of types , (m odd), and are also provided
Opposition diagrams for automorphisms of small spherical buildings
An automorphism of a spherical building is called
\textit{capped} if it satisfies the following property: if there exist both
type and simplices of mapped onto opposite simplices by
then there exists a type simplex of mapped onto
an opposite simplex by . In previous work we showed that if is
a thick irreducible spherical building of rank at least with no Fano plane
residues then every automorphism of is capped. In the present work we
consider the spherical buildings with Fano plane residues (the \textit{small
buildings}). We show that uncapped automorphisms exist in these buildings and
develop an enhanced notion of "opposition diagrams" to capture the structure of
these automorphisms. Moreover we provide applications to the theory of
"domesticity" in spherical buildings, including the complete classification of
domestic automorphisms of small buildings of types and
Opposition diagrams for automorphisms of large spherical buildings
Let be an automorphism of a thick irreducible spherical building
of rank at least with no Fano plane residues. We prove that if
there exist both type and simplices of mapped onto
opposite simplices by , then there exists a type simplex
of mapped onto an opposite simplex by . This property is
called "cappedness". We give applications of cappedness to opposition diagrams,
domesticity, and the calculation of displacement in spherical buildings. In a
companion piece to this paper we study the thick irreducible spherical
buildings containing Fano plane residues. In these buildings automorphisms are
not necessarily capped
Automorphisms and opposition in twin buildings
We show that every automorphism of a thick twin building interchanging the
halves of the building maps some residue to an opposite one. Furthermore we
show that no automorphism of a locally finite 2-spherical twin building of rank
at least 3 maps every residue of one fixed type to an opposite. The main
ingredient of the proof is a lemma that states that every duality of a thick
finite projective plane admits an absolute point, i.e., a point mapped onto an
incident line. Our results also hold for all finite irreducible spherical
buildings of rank at least 3, and as a consequence we deduce that every
involution of a thick irreducible finite spherical building of rank at least 3
has a fixed residue
Finiteness Properties of Chevalley Groups over the Laurent Polynomial Ring over a Finite Field
We show that if G is a Chevalley group of rank n and F_q[t,t^{-1}] is the
ring of Laurent polynomials over a finite field, then G(F_q[t,t^{-1}]) is of
type F_{2n-1}. This bound is optimal because it is known -- and we show again
-- that the group is not of type F_{2n}.Comment: 36 pages, 4 figure
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.Comment: To appear in Proc. London Math. So
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