Deformations of lie subgroups and the variation of isotropy subgroups

Equivalence among isotropy subgroups of space groups.

The Landau theory' of continuous phase transitions provides a powerful tool for understanding transitions between solid phases whose symmetries have a group-subgroup relationship. In the Landau theory, the thermodynamic free energy F of the crystal is written as a function of an order parameter Q. In the high-symmetry phase, the minimum of F is at Q = 0. In the low-symmetry phase, the minimum o...

A Note on Counting Isotropy Subgroups

Deficiencies of Lattice Subgroups of Lie Groups

Let Γ be a lattice in a connected Lie group. We show that besides a few exceptional cases, the deficiency of Γ is nonpositive.

Cocompact Subgroups of Semisimple Lie Groups

Lattices and parabolic subgroups are the obvious examples of cocompact subgroups of a connected, semisimple Lie group with finite center. We use an argument of C. C. Moore to show that every cocompact subgroup is, roughly speaking, a combination of these. We study a cocompact subgroup H of a connected, semisimple Lie group G with finite center. The case where H is discrete is very important and...

Superrigid Subgroups of Solvable Lie Groups

Let Γ be a discrete subgroup of a simply connected, solvable Lie group G, such that AdG Γ has the same Zariski closure as AdG. If α : Γ → GLn(R) is any finite-dimensional representation of Γ, we show that α virtually extends to a continuous representation σ of G. Furthermore, the image of σ is contained in the Zariski closure of the image of α. When Γ is not discrete, the same conclusions are t...