44 research outputs found

    Polynomial dynamic and lattice orbits in S-arithmetic homogeneous spaces

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    Consider an homogeneous space under a locally compact group G and a lattice in G. Then the lattice naturally acts on the homogeneous space. Looking at a dense orbit, one may wonder how to describe its repartition. One then adopt a dynamical point of view and compare the asymptotic distribution of points in the orbits with the natural measure on the space. In the setting of Lie groups and their homogeneous spaces, several results showed an equidistribution of points in the orbits using Ratner's rigidity of polynomial dynamics in homogeneous spaces. We address here this problem in the setting of p-adic and S-arithmetic groups

    On SL(3,C\mathbb C)-representations of the Whitehead link group

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    We describe a family of representations in SL(3,C\mathbb C) of the fundamental group π\pi of the Whitehead link complement. These representations are obtained by considering pairs of regular order three elements in SL(3,C\mathbb C) and can be seen as factorising through a quotient of π\pi defined by a certain exceptional Dehn surgery on the Whitehead link. Our main result is that these representations form an algebraic component of the SL(3,C\mathbb C)-character variety of π\pi.Comment: 20 pages, 3 figures, 4 tables, and a companion Sage notebook (see the references) v2: A few corrections and improvement

    Dimension of character varieties for 33-manifolds

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    Let MM be a 33-manifold, compact with boundary and Γ\Gamma its fundamental group. Consider a complex reductive algebraic group G. The character variety X(Γ,G)X(\Gamma,G) is the GIT quotient Hom(Γ,G)//G\mathrm{Hom}(\Gamma,G)//G of the space of morphisms Γ→G\Gamma\to G by the natural action by conjugation of GG. In the case G=SL(2,C)G=\mathrm{SL}(2,\mathbb C) this space has been thoroughly studied. Following work of Thurston, as presented by Culler-Shalen, we give a lower bound for the dimension of irreducible components of X(Γ,G)X(\Gamma,G) in terms of the Euler characteristic χ(M)\chi(M) of MM, the number tt of torus boundary components of MM, the dimension dd and the rank rr of GG. Indeed, under mild assumptions on an irreducible component X0X_0 of X(Γ,G)X(\Gamma,G), we prove the inequality dim(X0)≄t⋅r−dχ(M).\mathrm{dim}(X_0)\geq t \cdot r - d\chi(M).Comment: 12 pages, 1 figur

    A brief remark on orbits of SL(2,â„€) in the Euclidean plane

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    International audienceThe repartition of dense orbits of lattices in the Euclidean plane were described by Ledrappier and Nogueira. We present here an elementary description of the gaps appearing in the experimentations. The main idea behind this description is to see the Euclidean plane as the space of (upper triangular) unipotent orbits in SL(2,ℝ). We conclude with the remark that this analysis may be carried on in much more general settings

    DĂ©crire l'Ă©volution d'un nuage de points

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    International audienceNous expliquons sur un exemple les phénomÚnes de répartition asymptotique. En pratique, nous étudions l'ensemble des images d'un point fixé du plan sous l'action des matrices de SL(2,Z)SL(2,Z). Nous montrons que nous pouvons décrire la répartition de cet ensemble dans le plan

    SOUS-GROUPES H-LOXODROMIQUES

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    International audienceConsider k a finite extension of Q_p, with p a prime number. Let H be a finite index subgroup of k^* and G be the group SL(n,k) with its Zariski topology of Q _p-group. We investigate the existence of a subgroup of G which is Zariski-dense and such that each of its elements has a spectrum included in H. A necessary and sufficient condition is obtained: such a subgroup exists if and only if either -1 belongs to H or the dimension n is not congruent to 2 modulo 4.On considùre une extension finie k de Qp, avec p un nombre premier, H un sous-groupe d'indice fini de k * et le groupe SL(n, k). Nous montrons que SL(n, k) admet un sous-groupe Qp-Zariski-dense dont toutes les matrices ont leur spectre inclus dans H si et seulement si soit −1 est dans le sous-groupe H, soit n n'est pas congru à 2 modulo 4. Abstract (H-loxodromic subgroups). — Consider k a finite extension of Qp, with p a prime number. Let H be a finite index subgroup of k * and G be the group SL(n, k) with its Zariski topology of Qp-group. We investigate the existence of a subgroup of G which is Zariski-dense and such that each of its elements has a spectrum included in H. A necessary and sufficient condition is obtained: such a subgroup exists if and only if either −1 belongs to H or the dimension n is not congruent to 2 modulo 4
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