On discrete Zariski-dense subgroups of Algebraic Groups

Jörg Winkelmann

Let k be a locally compact field, i.e. a field equipped with an absolute value inducing a non-discrete locally compact topology. (Every locally compact field is isomorphic to C, R, a finite extension of a field Q_p or F_q((t)), where F_q is a finite field.)

We investigate under which conditions an algebraic group G defined over such a locally compact field k admits a subgroup which is a discrete subgroup of G(k) in the topology induced by the locally compact topology on k, but dense with respect to the Zariski topology.

A main motivation for our investigation was the following. A discrete cocompact subgroup of an algebraic group defined over C is necessarily Zariski-dense. There are known obstructions to the existence of discrete cocompact subgroups, e.g. only unimodular groups may admit discrete cocompact subgroups. Thus one may ask whether these obstructions actually prohibit only discrete cocompact subgroups or prohibit all Zariski-dense discrete subgroups. This is not the case. It turns out that in most cases it is much easier to find Zariski-dense discrete subgroups than it is to find lattices. In particular there are many non-unimodular groups containing Zariski-dense discrete subgroups although in some special cases unimodularity is necessary for the existence of Zariski-dense discrete subgroups.

For Zariski-connected non-solvable groups we completely characterize those k-groups which contain Zariski-dense discrete subgroups. In the case where the radical R of G is defined over k our result takes the following form:

The following conditions are equivalent:

• G(k) contains a free discrete Zariski-dense subgroup.
• G(k) contains a discrete Zariski-dense subgroup.
• G(k) is not amenable.
• The image of G(k) in (G/R)(k) is not relatively compact.

• G(k)/R(k) is not compact.
• G/R is k-isotropic.

For solvable k-groups we have such a complete description only in the case of p-adic fields.

Borel subgroups in simple k-groups never contain Zariski-dense discrete subgroups. In contrast, for some locally compact fields (k=C or char(k)>0) the Borel k-subgroups of (SL₂)ⁿ (n>1) contain Zariski-dense discrete subgroups.

Our further results on solvable k-groups demonstrate that in the solvable case the picture depends heavily on the type of the local field in question. For instance, unipotent groups over p-adic fields never contain non-trivial discrete subgroups. Malcev proved that a unipotent R-group contains a Zariski-dense discrete subgroup if and only if it can be defined over Q. For locally compact fields of positive characteristics we prove that every k-split unipotent k-group contains a Zariski-dense discrete subgroup.

Math. Nachrichten 186 , 285 - 302 (1997)