Let P be a symplectic manifold with a free symplectic action of a connected compact Lie group G. We show that, given a certain transversality condition, the set of relative equilibria E nearpe ∈ E of aG-invariant Hamiltonian system onP is locally Whitney-stratified by the conjugacy classes of the isotropy subgroups (under the product of the coadjoint and adjoint actions) of the momentum-generat...

We have proved in [Pr] that fundamental groups of oriented geometrizable 3manifolds have a solvable conjugacy problem. We now focus on groups of non-oriented geometrizable 3-manifolds in order to conclude that all groups of geometrizable 3-manifolds have a solvable conjugacy problem.

We use wreath products to provide criteria for a group to be conjugacy separable or omnipotent. These criteria are in terms of virtual retractions onto cyclic subgroups. We give two applications: a straightforward topological proof of the theorem of Stebe that infiniteorder elements of Fuchsian groups (of the first type) are conjugacy distinguished, and a proof that surface groups are omnipotent.

We show that there exist a finite number of Cartan subgroups up to conjugacy in SL2(Qp) and we describe all of them. We show that the Cartan subgroup consisting of all diagonal matrices is generous and it is the only one up to conjugacy.

An S3-involution graph for a group G is a graph with vertex set a union of conjugacy classes of involutions of G such that two involutions are adjacent if they generate an S3-subgroup in a particular set of conjugacy classes. We investigate such graphs in general and also for the case where G = PSL(2, q).

It is well known that the dynamics and conjugacy class of a complex Möbius transformation can be determined from a simple rational function of the coefficients of the transformation. We study the group of quaternionic Möbius transformations and identify simple rational functions of the coefficients of the transformations that determine dynamics and conjugacy.

An S3-involution graph for a group G is a graph with vertex set a union of conjugacy classes of involutions of G such that two involutions are adjacent if they generate an S3-subgroup in a particular set of conjugacy classes. We investigate such graphs in general and also for the case where G = PSL(2, q). 2000 Mathematics subject classification: 20B25, 05C25.

We use combinatorial techniques to derive explicit formulas for the number of conjugacy classes of elements of finite order in unitary, symplectic, and orthogonal Lie groups, as well as the number of such conjugacy classes whose elements have a specified number of distinct eigenvalues.

We construct a hyperbolic group containing a finitely presented subgroup, which has infinitely many conjugacy classes of finite-order elements. We also use a version of Morse theory with high dimensional horizontal cells and use handle cancellation arguments to produce other examples of subgroups of CAT(0) groups with infinitely many conjugacy classes of finite-order elements.

Let G be a semisimple, simply connected, algebraic group over an algebraically closed field k with Lie algebra g. We study the spaces of parahoric subalgebras of a given type containing a fixed nil-elliptic element of g⊗k((π)), i.e. fixed point varieties on affine flag manifolds. We define a natural class of k-actions on affine flag manifolds, generalizing actions introduced by Lusztig and Smel...