2,413 research outputs found
Spherical subcomplexes of spherical buildings
Let B be a thick spherical building equipped with its natural CAT(1) metric
and let M be a proper, convex subset of B. If M is open or if M is a closed
ball of radius pi/2, then the maximal subcomplex supported by the complement of
M is spherical and non contractible.Comment: 33 pages, 7 figure
Deformation and rigidity of simplicial group actions on trees
We study a notion of deformation for simplicial trees with group actions
(G-trees). Here G is a fixed, arbitrary group. Two G-trees are related by a
deformation if there is a finite sequence of collapse and expansion moves
joining them. We show that this relation on the set of G-trees has several
characterizations, in terms of dynamics, coarse geometry, and length functions.
Next we study the deformation space of a fixed G-tree X. We show that if X is
`strongly slide-free' then it is the unique reduced tree in its deformation
space.
These methods allow us to extend the rigidity theorem of Bass and Lubotzky to
trees that are not locally finite. This yields a unique factorization theorem
for certain graphs of groups. We apply the theory to generalized
Baumslag-Solitar groups and show that many have canonical decompositions. We
also prove a quasi-isometric rigidity theorem for strongly slide-free G-trees.Comment: Published in Geometry and Topology at
http://www.maths.warwick.ac.uk/gt/GTVol6/paper8.abs.htm
A (very short) introduction to buildings
These lectures are an informal elementary introduction to buildings. They are
written for, and by, a non-expert. The aim is to get to the definition of a
building and feel that it is an entirely natural thing. To maintain the lecture
style examples have replaced proofs. The notes at the end indicate where these
proofs can be found. Most of what we say has its origins in the work of Jacques
Tits, and our account borrows heavily from the books of Abramenko and Brown and
of Ronan. Lecture 1 illustrates all the essential features of a building in the
context of an example, but without mentioning any building terminology. In
principle anyone could read this. Lectures 2-4 firm-up and generalize these
specifics: Coxeter groups appear in Lecture 2, chambers systems in Lecture 3
and the definition of a building in Lecture 4. Lecture 5 addresses where
buildings come from by describing the first important example: the spherical
building of an algebraic group.Comment: Minor improvements and some new pictures; 23 page
Presentations for the punctured mapping class groups in terms of Artin groups
Consider an oriented compact surface F of positive genus, possibly with
boundary, and a finite set P of punctures in the interior of F, and define the
punctured mapping class group of F relatively to P to be the group of isotopy
classes of orientation-preserving homeomorphisms h: F-->F which pointwise fix
the boundary of F and such that h(P) = P. In this paper, we calculate
presentations for all punctured mapping class groups. More precisely, we show
that these groups are isomorphic with quotients of Artin groups by some
relations involving fundamental elements of parabolic subgroups.Comment: Published by Algebraic and Geometric Topology at
http://www.maths.warwick.ac.uk/agt/AGTVol1/agt-1-5.abs.htm
Richardson Varieties Have Kawamata Log Terminal Singularities
Let be a Richardson variety in the full flag variety associated
to a symmetrizable Kac-Moody group . Recall that is the intersection
of the finite dimensional Schubert variety with the finite codimensional
opposite Schubert variety . We give an explicit \bQ-divisor on
and prove that the pair has Kawamata log terminal
singularities. In fact, is ample, which additionally
proves that is log Fano.
We first give a proof of our result in the finite case (i.e., in the case
when is a finite dimensional semisimple group) by a careful analysis of an
explicit resolution of singularities of (similar to the BSDH resolution
of the Schubert varieties). In the general Kac-Moody case, in the absence of an
explicit resolution of as above, we give a proof that relies on the
Frobenius splitting methods. In particular, we use Mathieu's result asserting
that the Richardson varieties are Frobenius split, and combine it with a result
of N. Hara and K.-I. Watanabe relating Frobenius splittings with log canonical
singularities.Comment: 15 pages, improved exposition and explanation. To appear in the
International Mathematics Research Notice
A Quasi Curtis-Tits-Phan theorem for the symplectic group
We obtain the symplectic group \SP(V) as the universal completion of an
amalgam of low rank subgroups akin to Levi components. We let \SP(V) act
flag-transitively on the geometry of maximal rank subspaces of . We show
that this geometry and its rank residues are simply connected with few
exceptions. The main exceptional residue is described in some detail. The
amalgamation result is then obtained by applying Tits' lemma. This provides a
new way of recognizing the symplectic groups from a small collection of small
subgroups
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