9,908 research outputs found
Convex cocompact actions in real projective geometry
We study a notion of convex cocompactness for (not necessarily irreducible)
discrete subgroups of the projective general linear group acting on real
projective space, and give various characterizations. A convex cocompact group
in this sense need not be word hyperbolic, but we show that it still has some
of the good properties of classical convex cocompact subgroups in rank-one Lie
groups. Extending our earlier work arXiv:1701.09136 from the context of
projective orthogonal groups, we show that for word hyperbolic groups
preserving a properly convex open set in projective space, the above general
notion of convex cocompactness is equivalent to a stronger convex cocompactness
condition studied by Crampon-Marquis, and also to the condition that the
natural inclusion be a projective Anosov representation. We investigate
examples.Comment: 77 pages, 6 figures. Added appendix. Removed section on Anosov
right-angled reflection groups, which will appear as a separate pape
Coxeter group in Hilbert geometry
A theorem of Tits - Vinberg allows to build an action of a Coxeter group
on a properly convex open set of the real projective space,
thanks to the data of a polytope and reflection across its facets. We give
sufficient conditions for such action to be of finite covolume,
convex-cocompact or geometrically finite. We describe an hypothesis that make
those conditions necessary. Under this hypothesis, we describe the Zariski
closure of , find the maximal -invariant convex, when there is
a unique -invariant convex, when the convex is strictly
convex, when we can find a -invariant convex which is
strictly convex.Comment: 48
Algebraic curves with many automorphisms
Let be a (projective, geometrically irreducible, nonsingular) algebraic
curve of genus defined over an algebraically closed field of odd
characteristic . Let be the group of all automorphisms of which
fix element-wise. It is known that if then the -rank
(equivalently, the Hasse-Witt invariant) of is zero. This raises the
problem of determining the (minimum-value) function such that whenever
then has zero -rank. For {\em{even}} we prove
that . The {\em{odd}} genus case appears to be much more
difficult although, for any genus , if has a solvable
subgroup such that then has zero -rank and fixes a
point of . Our proofs use the Hurwitz genus formula and the Deuring
Shafarevich formula together with a few deep results from finite group theory
characterizing finite simple groups whose Sylow -subgroups have a cyclic
subgroup of index . We also point out some connections with the Abhyankar
conjecture and the Katz-Gabber covers
Period three actions on lens spaces
We show that a free period three action on a lens space is standard, i.e. the
quotient is homeomorphic to a lens space. This is an extension of the result
for period three actions on the three-sphere, arXiv:math.GT/0204077, by the
author and J. Hyam Rubinstein.Comment: 67 pages, 54 picture
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