9,908 research outputs found

    Convex cocompact actions in real projective geometry

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    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

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    A theorem of Tits - Vinberg allows to build an action of a Coxeter group Γ\Gamma on a properly convex open set Ω\Omega of the real projective space, thanks to the data PP 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 Γ\Gamma, find the maximal Γ\Gamma-invariant convex, when there is a unique Γ\Gamma-invariant convex, when the convex Ω\Omega is strictly convex, when we can find a Γ\Gamma-invariant convex Ω′\Omega' which is strictly convex.Comment: 48

    Algebraic curves with many automorphisms

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    Let XX be a (projective, geometrically irreducible, nonsingular) algebraic curve of genus g≥2g \ge 2 defined over an algebraically closed field KK of odd characteristic pp. Let Aut(X)Aut(X) be the group of all automorphisms of XX which fix KK element-wise. It is known that if ∣Aut(X)∣≥8g3|Aut(X)|\geq 8g^3 then the pp-rank (equivalently, the Hasse-Witt invariant) of XX is zero. This raises the problem of determining the (minimum-value) function f(g)f(g) such that whenever ∣Aut(X)∣≥f(g)|Aut(X)|\geq f(g) then XX has zero pp-rank. For {\em{even}} gg we prove that f(g)≤900g2f(g)\leq 900 g^2. The {\em{odd}} genus case appears to be much more difficult although, for any genus g≥2g\geq 2, if Aut(X)Aut(X) has a solvable subgroup GG such that ∣G∣>252g2|G|>252 g^2 then XX has zero pp-rank and GG fixes a point of XX. 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 22-subgroups have a cyclic subgroup of index 22. We also point out some connections with the Abhyankar conjecture and the Katz-Gabber covers

    Period three actions on lens spaces

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    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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