Weakly Commensurable Zariski-Dense Subgroups of Algebraic Groups Defined Over Fields of Positive Characteristic

In [PR09], Prasad and Rapinchuk introduce and analyze a new relationship, called `weak commensurability,' between (Zariski-dense) abstract subgroups of the groups of $K$-rational points of connected semisimple algebraic groups. Numerous results have been shown for algebraic groups defined over fields of characteristic zero, but not for fields of positive characteristic. The main purpose of this work is to extend the notion of weak commensurability to fields of positive characteristic, specifically to prove and analyze characteristic $p>0$ analogs of the results from~\cite{PR09}.

We develop several characteristic $p>0$ results showing that weakly commensurable Zariski-dense subgroups must share structural properties. Specifically, we show that the Zariski-closure of two weakly commensurable Zariski-dense subgroups of absolutely almost simple groups must have the same Killing-Cartan type. The trace field of a subgroup is the field generated by $1$ and the traces of elements in the Zariski-dense subgroup. We show that a $p$th power of the trace field of one Zariski-dense subgroup contained in the trace field of a weakly commensurable subgroup. We also prove a similar statement when we replace the trace field with the minimal Galois extension of the trace field such that the Zariski-closure of the subgroup is an inner form over this field. We also show that discreteness is a property that is shared by weakly commensurable subgroups.