44 research outputs found

### Polynomial dynamic and lattice orbits in S-arithmetic homogeneous spaces

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,$\mathbb C$)-representations of the Whitehead link group

We describe a family of representations in SL(3,$\mathbb C$) of the
fundamental group $\pi$ of the Whitehead link complement. These representations
are obtained by considering pairs of regular order three elements in
SL(3,$\mathbb C$) and can be seen as factorising through a quotient of $\pi$
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,$\mathbb C$)-character variety of $\pi$.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 $3$-manifolds

Let $M$ be a $3$-manifold, compact with boundary and $\Gamma$ its fundamental
group. Consider a complex reductive algebraic group G. The character variety
$X(\Gamma,G)$ is the GIT quotient $\mathrm{Hom}(\Gamma,G)//G$ of the space of
morphisms $\Gamma\to G$ by the natural action by conjugation of $G$. In the
case $G=\mathrm{SL}(2,\mathbb C)$ 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 $X(\Gamma,G)$ in terms of
the Euler characteristic $\chi(M)$ of $M$, the number $t$ of torus boundary
components of $M$, the dimension $d$ and the rank $r$ of $G$. Indeed, under
mild assumptions on an irreducible component $X_0$ of $X(\Gamma,G)$, we prove
the inequality $\mathrm{dim}(X_0)\geq t \cdot r - d\chi(M).$Comment: 12 pages, 1 figur

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

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,â). 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

International audienceNous expliquons sur un exemple les phĂ©nomĂšnes de rĂ©partition asymptotique. En pratique, nous Ă©tudions l'ensemble des images d'un point fixĂ© du plan sous l'action des matrices de $SL(2,Z)$. Nous montrons que nous pouvons dĂ©crire la rĂ©partition de cet ensemble dans le plan

### SOUS-GROUPES H-LOXODROMIQUES

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 considĂšre 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 â1 est dans le sous-groupe H, soit n n'est pas congru Ă 2 modulo 4. Abstract (H-loxodromic subgroups). â 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 â1 belongs to H or the dimension n is not congruent to 2 modulo 4