## Asymptotic properties of linear groups. (Propriétés asymptotiques des groupes linéaires.)(French)Zbl 0947.22003

This paper studies asymptotic properties of elements of Zariski dense subgroups of reductive linear Lie groups, after making a reasonable projection of all the elements into a fundamental domain for the diagonalizable elements under conjugation. After this projection, there are two sets that might be called the “asymptotic closed cone” (my choice of words) of $$\Gamma$$. These two candidates are shown to be the same. More specifically: Let $$\{G\}$$ be a reductive linear real Lie group, let $$G$$ denote the real points of $$\{G\}$$ and let $$\Gamma$$ be a Zariski dense subgroup of $$\{G\}$$. Let $$A^+$$ denote a positive Weyl chamber for $$G$$ with respect to some maximal split torus. If $$K$$ is a well-chosen maximal compact subgroup of $$G$$, then each element $$g$$ of $$G$$ may be written $$g=kak'$$, where $$k,k'\in K$$ and where $$a\in A^+$$. The map $$\mu :G\to A^+$$ is defined by $$\mu(g) =a$$.
We will say that $$v\in{\mathfrak a} ^+$$ is asymptotic if there is a sequence $$\gamma_i$$ in $$\Gamma$$ such that $$\gamma _i\to\infty$$ in $$G$$ and such that $$\mathbb{R}\log (\mu(\gamma _i))$$ converges to $$\mathbb{R} v$$ in the space of lines in $${\mathfrak a}$$. The convex hull of all such asymptotic vectors $$v$$ is called the “asymptotic cone of $$\log (\mu (\Gamma))$$”.
For every element $$g\in G$$, there are unique pairwise-commuting elements $$g_h,g_e,g_u\in G$$ such that $$g=g_hg_eg_u$$, such that $$g_h$$ is diagonalizable over $$\mathbb{R}$$, such that $$g_e$$ lies in a compact subgroup of $$G$$ and such that $$g_u$$ is unipotent.
Define $$\Lambda :G\to A^+$$ by letting $$\lambda(g)$$ be the unique element of $$A^+$$ which is conjugate to $$g_h$$. Let $$l_\Gamma$$ denote the closure of the convex hull of the union of all lines $$\mathbb{R}\log (\lambda (\gamma))$$, where the union is over all $$\gamma\in\Gamma$$.
The main result of the paper asserts that the asymptotic cone of $$\log(\mu (\Gamma))$$ is $$l_\Gamma$$. A converse is also proved and generalizations are obtained to other local fields besides $$\mathbb{R}$$.

### MathOverflow Questions:

Spectrum of random matrices has dominant eigenvalues

### MSC:

 2.2e+16 General properties and structure of real Lie groups
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