Asymptotic properties of linear groups.
(Propriétés asymptotiques des groupes linéaires.)

*(French)*Zbl 0947.22003This 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 \(\mathbb{R}\).

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 \(\mathbb{R}\).

Reviewer: Scot Adams (MR 98b:22010)

##### MSC:

22E15 | General properties and structure of real Lie groups |