166 research outputs found
A question of Frohardt on -groups, and skew translation quadrangles of even order
We solve a fundamental question posed in Frohardt's 1988 paper [Fro] on
finite -groups with Kantor familes, by showing that finite groups with a
Kantor family having distinct members such that is a central subgroup of and the
quotient is abelian cannot exist if the center of has
exponent and the members of are elementary abelian. In a
similar way, we solve another old problem dating back to the 1970s by showing
that finite skew translation quadrangles of even order are always
translation generalized quadrangles.Comment: 10 pages; submitted (February 2018
Central aspects of skew translation quadrangles, I
Except for the Hermitian buildings , up to a combination
of duality, translation duality or Payne integration, every known finite
building of type satisfies a set of general synthetic
properties, usually put together in the term "skew translation generalized
quadrangle" (STGQ). In this series of papers, we classify finite skew
translation generalized quadrangles. In the first installment of the series, as
corollaries of the machinery we develop in the present paper, (a) we obtain the
surprising result that any skew translation quadrangle of odd order is
a symplectic quadrangle; (b) we determine all skew translation quadrangles with
distinct elation groups (a problem posed by Payne in a less general setting);
(c) we develop a structure theory for root-elations of skew translation
quadrangles which will also be used in further parts, and which essentially
tells us that a very general class of skew translation quadrangles admits the
theoretical maximal number of root-elations for each member, and hence all
members are "central" (the main property needed to control STGQs, as which will
be shown throughout); (d) we solve the Main Parameter Conjecture for a class of
STGQs containing the class of the previous item, and which conjecturally
coincides with the class of all STGQs.Comment: 66 pages; submitted (December 2013
Generalized polygons with non-discrete valuation defined by two-dimensional affine R-buildings
In this paper, we show that the building at infinity of a two-dimensional
affine R-building is a generalized polygon endowed with a valuation satisfying
some specific axioms. Specializing to the discrete case of affine buildings,
this solves part of a long standing conjecture about affine buildings of type
G~_2, and it reproves the results obtained mainly by the second author for
types A~_2 and C~_2. The techniques are completely different from the ones
employed in the discrete case, but they are considerably shorter, and general
(i.e., independent of the type of the two-dimensional R-building)
A geometric construction of panel-regular lattices in buildings of types ~A_2 and ~C_2
Using Singer polygons, we construct locally finite affine buildings of types
~A_2 and ~C_2 which admit uniform lattices acting regularly on panels. This
construction produces very explicit descriptions of these buildings as well as
very short presentations of the lattices. All but one of the ~C_2-buildings are
necessarily exotic. To the knowledge of the author, these are the first
presentations of lattices in buildings of type ~C_2. Integral and rational
group homology for the lattices is also calculated.Comment: 42 pages, small corrections and cleanup. Results are unchanged
A sixteen-relator presentation of an infinite hyperbolic Kazhdan group
We provide an explicit presentation of an infinite hyperbolic Kazhdan group
with generators and relators of length at most . That group acts
properly and cocompactly on a hyperbolic triangle building of type .
We also point out a variation of the construction that yields examples of
lattices in -buildings admitting non-Desarguesian residues of
arbitrary prime power order.Comment: 9 pages, 1 figur
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