n previous work, the author and his collaborators developed a relationship in the SL(2,C) representation theories of two finitely generated groups with isomorphicprofinite completions assuming a certain strong representation rigidity for one of thegroups. This was then exploited as one part of producing examples of lattices in SL(2,C) which are profinitely rigid. In this article, the relationship is extended to representations in any connected reductive algebraic groups under a weaker representation rigidity hypothesis. The results are applied to lattices in higher rank Liegroups where we show that for some such groups, including SL(n,Z) forn≥3, they are either profinitely rigid, or they contain a proper Grothendieck subgroup.

Extending an explicit result from Bridson–Conder–Reid, this work provides an algorithm for distinguishing finite quotients between cocompact triangle groups Δ ?and lattices Γ of constant curvature symmetric 2-spaces. Much of our attention will be on when these lattices are Fuchsian groups. We prove that it will suffice to take a finite quotient that is Abelian, dihedral, a subgroup of PSL(<i>n</i>,<b>F</b>_<i>q</i>) (for an odd prime power q), or an Abelian extension of one of these 3 groups. For the latter case, we will require and develop an approach for creating group extensions upon a shared finite quotient of Δ? and Γ which between them have differing degrees of smoothness. Furthermore, on the order of a finite quotient that distinguishes between ?Δ and Γ, we are able to establish an effective upperbound that is superexponential depending on the cone orders appearing in each group.<br>