Varieties of elements of given order in simple algebraic groups

Given a positive integer u and a simple algebraic group G defined over an algebraically closed field K of characteristic p, we derive properties about the subvariety G[u] of G consisting of elements of G of order dividing u. In particular, we determine the dimension of G[u], completing results of Lawther [7] in the special case where G is of adjoint type. We also apply our results to the study of finite simple quotients of triangle groups, giving further insight on a conjecture we proposed in [10] as well as proving that some finite quasisimple groups are not quotients of certain triangle groups.