Transformation groups with orbits of uniform dimension.

Groups synchronizing a transformation of non-uniform kernel

Suppose that a deterministic finite automata A = (Q,⌃) is such that all but one letters from the alphabet ⌃ act as permutations of the state set Q and the exceptional letter acts as a transformation with non-uniform kernel. Which properties of the permutation group G generated by the letters acting as permutations ensure that A becomes a synchronizing automaton under every possible choice of th...

Group Membership for Groups with Primitive Orbits

This paper considers a permutation group G = 〈S〉 of degree n with t orbits such that the action on each orbit is primitive. It presents a O(tn2 logc(n)) time Monte Carlo group membership algorithm for some constant c. The algorithm is notable for its use of a new theorem showing how to find O(t log n) generators in O (̃|S|n) time under a more general form of the above hypotheses. The algorithm r...

Uniform distribution of orbits of lattices

Complex Orbits of Solvable Groups

The following structure theorems are proved: An orbit of a real solvable Lie group in projective space that is a complex submanifold is isomorphic to C x (C*)m x Í2 , where Í2 is an open orbit of a real solvable Lie group in a projective rational variety. Also, any homogeneous space of a complex Lie group that is isomorphic to C" can be realized as an orbit in some projective space. As a conseq...

Distinguished Orbits of Reductive Groups

We prove a generalization and give a new proof of a theorem of Borel-Harish-Chandra on closed orbits of linear actions of reductive groups. Consider a real reductive algebraic group G acting linearly and rationally on a real vector space V . G can be viewed as the real points of a complex reductive group G C which acts on V C := V ⊗ C. In [BHC62] it was shown that G · v ∩ V is a finite union of...