19 research outputs found

### Proper affine actions on semisimple Lie algebras

For any noncompact semisimple real Lie group $G$, we construct a group of
affine transformations of its Lie algebra $\mathfrak{g}$ whose linear part is
Zariski-dense in $\operatorname{Ad} G$ and which is free, nonabelian and acts
properly discontinuously on $\mathfrak{g}$.Comment: Section 3 of this paper draws heavily on the section 3 from my
earlier paper arXiv:1303.3766 This is the version that appeared in Geometriae
Dedicata. I have corrected some mistakes, added a few examples and added a
proof that the Margulis invariant is well-define

### Action of $w_0$ on $V^L$: the special case of $\mathfrak{so}(1,n)$

In this note, we present an algorithm that allows to answer any individual
instance of the following question. Let $G_{\mathbb{R}}$ be a semisimple real
Lie group, and $V$ an irreducible representation of $G_{\mathbb{R}}$. How does
the longest element $w_0$ of the restricted Weyl group $W$ act on the subspace
$V^L$ of $V$ formed by vectors that are invariant by $L$, the centralizer of a
maximal split torus of $G_{\mathbb{R}}$? This algorithm comprises two parts.
First we describe a complete answer to this question in the particular case
where $G_{\mathbb{R}} = \operatorname{SO}(1,n)$ for any $n \geq 2$. Then, for
an arbitrary $G_{\mathbb{R}}$, we show that it suffices to do the computation
in a well-chosen subgroup $S_{\mathbb{R}} \subset G_{\mathbb{R}}$ which is (up
to isogeny) the product of several groups that are either compact, abelian or
isomorphic to $\operatorname{SO}(1,n)$ for some $n \geq 2$.Comment: Main text: 9 pages, 1 figure. Appendix: 7 pages, of which 4.5 pages
are occupied by 2 tables. The ancillary files contain an implementation of
the algorithm in the LiE software package (with an explanation

### New sequences of non-free rational points

We exhibit some new infinite families of rational values of $\tau$, some of
them squares of rationals, for which the group or even the semigroup generated
by the matrices $\left( \begin{smallmatrix} 1 & 1 \\ 0 & 1 \end{smallmatrix}
\right)$ and $\left( \begin{smallmatrix} 1 & 0 \\ \tau & 1 \end{smallmatrix}
\right)$ is not free.Comment: Fixed a typo: I had written "free" instead of "non-free" in a few
places; and made a few other minor edit

### New sequences of non-free rational points

We exhibit some new infinite families of rational values of $\tau$, some of them squares of rationals, for which the group or even the semigroup generated by the matrices $({{\textstyle \begin{matrix} 1 & 1\\ 0 & 1 \end{matrix}}})$ and $({{\textstyle \begin{matrix} 1 & 0 \\ \tau & 1 \end{matrix}}})$ is not free