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European Mathematical Society Publishing House
2024-03-28 13:53:36
5
https://www.ems-ph.org/meta/jmeta-stream.php?jrn=GGD&vol=6&iss=3&update_since=2024-03-28
Groups, Geometry, and Dynamics
Groups Geom. Dyn.
GGD
1661-7207
1661-7215
Group theory and generalizations
10.4171/GGD
http://www.ems-ph.org/doi/10.4171/GGD
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European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society
6
2012
3
On the surjectivity of Engel words on PSL(2,q)
Tatiana
Bandman
Bar-Ilan University, RAMAT GAN, ISRAEL
Shelly
Garion
Universität Münster, MÜNSTER, GERMANY
Fritz
Grunewald
Heinrich-Heine-Universität, DÜSSELDORF, GERMANY
Engel words, special linear group, arithmetic dynamics, periodic points, finite fields, trace map
We investigate the surjectivity of the word map defined by the $n$-th Engel word on the groups $\mathrm{PSL}(2,q)$ and $\mathrm{SL}(2,q)$. For $\mathrm{SL}(2,q)$ we show that this map is surjective onto the subset $\mathrm{SL}(2,q)\setminus\{-\mathrm{id}\}\subset \mathrm{SL}(2,q)$ provided that $q \geq q_0(n)$ is sufficiently large. Moreover, we give an estimate for $q_0(n)$. We also present examples demonstrating that this does not hold for all $q$. We conclude that the $n$-th Engel word map is surjective for the groups $\mathrm{PSL}(2,q)$ when $q \geq q_0(n)$. By using a computer, we sharpen this result and show that for any $n \leq 4$ the corresponding map is surjective for all the groups $\mathrm{PSL}(2,q)$. This provides evidence for a conjecture of Shalev regarding Engel words in finite simple groups. In addition, we show that the $n$-th Engel word map is almost measure-preserving for the family of groups $\mathrm{PSL}(2,q)$, with $q$ odd, answering another question of Shalev. Our techniques are based on the method developed by Bandman, Grunewald and Kunyavskii for verbal dynamical systems in the group $\mathrm{SL}(2,q)$.
Algebraic geometry
Group theory and generalizations
Dynamical systems and ergodic theory
General
409
439
10.4171/GGD/162
http://www.ems-ph.org/doi/10.4171/GGD/162
Anosov AdS representations are quasi-Fuchsian
Quentin
Mérigot
Université Joseph Fourier, GRENOBLE CEDEX 9, FRANCE
Thierry
Barbot
Université d'Avignon, AVIGNON, FRANCE
Globally hyperbolic AdS spacetimes, Anosov representations
Let $\Gamma$ be a cocompact lattice in $\mathrm{SO}(1,n)$. A representation $\rho\colon \Gamma \to \mathrm{SO}(2,n)$ is called quasi-Fuchsian if it is faithful, discrete, and preserves an acausal subset in the boundary of anti-de Sitter space. A special case are Fuchsian representations, i.e., compositions of the inclusions $\Gamma \subset \mathrm{SO}(1,n)$ and $\mathrm{SO}(1,n) \subset \mathrm{SO}(2,n)$. We prove that quasi-Fuchsian representations are precisely those representations which are Anosov in the sense of Labourie (cf. (Lab06]). The study involves the geometry of locally anti-de Sitter spaces: quasi-Fuchsian representations are holonomy representations of globally hyperbolic spacetimes diffeomorphic to $\mathbb{R} \times \Gamma\backslash\mathbb{H}^{n}$ locally modeled on $\mathrm{AdS}_{n+1}$.
Differential geometry
Group theory and generalizations
General
441
483
10.4171/GGD/163
http://www.ems-ph.org/doi/10.4171/GGD/163
Cohomology computations for Artin groups, Bestvina–Brady groups, and graph products
Michael
Davis
Ohio State University, COLUMBUS, UNITED STATES
Boris
Okun
University of Wisconsin at Milwaukee, MILWAUKEE, UNITED STATES
Artin group, Bestvina–Brady group, building, Coxeter group, graph product, right-angled Artin group, $L^2$-Betti number, weighted $L^2$-cohomology
We compute: the cohomology with group ring coefficients of Artin groups (or actually, of their associated Salvetti complexes), of Bestvina–Brady groups of type FP, and of graph products of groups, the $L^2$-Betti numbers of Bestvina–Brady groups of type FP over $\mathbb{Q}$, and of graph products of groups, the weighted $L^2$-Betti numbers of graph products of Coxeter groups. In the case of arbitrary graph products there is an additional proviso: either all factors are infinite or all are finite.
Group theory and generalizations
General
485
531
10.4171/GGD/164
http://www.ems-ph.org/doi/10.4171/GGD/164
Limits of Baumslag–Solitar groups and dimension estimates in the space of marked groups
Luc
Guyot
Universität Göttingen, GÖTTINGEN, GERMANY
Yves
Stalder
Université Blaise Pascal, AUBIÈRE CEDEX, FRANCE
Baumslag–Solitar groups, space of marked groups, Turing degree, Hausdorff dimension
We prove that the limits of Baumslag–Solitar groups studied by the authors are non-linear hopfian C*-simple groups with infinitely many twisted conjugacy classes. We exhibit infinite presentations for these groups, classify them up to group isomorphism, describe their automorphisms and discuss the word and conjugacy problems. Finally, we prove that the set of these groups has non-zero Hausdorff dimension in the space of marked groups on two generators.
Group theory and generalizations
General
533
577
10.4171/GGD/165
http://www.ems-ph.org/doi/10.4171/GGD/165
Isometry groups of proper CAT(0)-spaces of rank one
Ursula
Hamenstädt
Universität Bonn, BONN, GERMANY
Bounded cohomology, isometry groups, CAT(0)-spaces, rigidity
Let $X$ be a proper CAT(0)-space and let $G$ be a closed subgroup of the isometry group $\mathrm{Iso}(X)$ of $X$. We show that if $G$ is non-elementary and contains a rank-one element then its second continuous bounded cohomology group with coefficients in the regular representation is non-trivial. As a consequence, up to passing to an open subgroup of finite index, either $G$ is a compact extension of a totally disconnected group or $G$ is a compact extension of a simple Lie group of rank one.
Group theory and generalizations
General
579
618
10.4171/GGD/166
http://www.ems-ph.org/doi/10.4171/GGD/166