A Note on Counting Isotropy Subgroups

A Note on Counting 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 Cuspidal

For geometrically nite Kleinian groups with parabolic elements we study that part of the Lagrange spectrum which does not lie in the Markov spectrum. Using the ergodicity of the associated geodesic ow with respect to the Liouville{Patterson measure, we obtain an estimate for the asymptotic frequency with which recurrent geodesics enter certain cusp regions. In particular, this allows a quantita...

A general algorithm for generating isotropy subgroups in superspace.

This paper presents a general algorithm for generating the isotropy subgroups of superspace extensions of crystallographic space groups involving arbitrary superpositions of multi-k order parameters from incommensurate and commensurate k vectors. Several examples are presented in detail in order to illuminate each step of the algorithm. The practical outcome is that one can now start with any c...

Counting Congruence Subgroups

Abstract. Let Γ denote the modular group SL(2, Z) and Cn(Γ) the number of congruence subgroups of Γ of index at most n. We prove that lim n→∞ log Cn(Γ) (log n)2/ log log n = 3−2 √ 2 4 . We also present a very general conjecture giving an asymptotic estimate for Cn(Γ) for general arithmetic groups. The lower bound of the conjecture is proved modulo the generalized Riemann hypothesis for Artin-He...