7,432 research outputs found
Geometry applications of irreducible representations of Lie Groups
In this note we give proofs of the following three algebraic facts which have applications in the theory of holonomy groups and homogeneous spaces: Any irreducibly acting connected subgroup G \subset Gl(n,\rr) is closed. Moreover, if admits an invariant bilinear form of Lorentzian signature, is maximal, i.e. it is conjugated to . We calculate the vector space of -invariant symmetric bilinear forms, show that it is at most -dimensional, and determine the maximal stabilizers for each dimension. Finally, we give some applications and present some open problem
Subgroup growth of lattices in semisimple Lie groups
We give very precise bounds for the congruence subgroup growth of arithmetic
groups. This allows us to determine the subgroup growth of irreducible lattices
of semisimple Lie groups. In the most general case our results depend on the
Generalized Riemann Hypothesis for number fields but we can state the following
unconditional theorem:
Let be a simple Lie group of real rank at least 2, different than
D_4(\bbc), and let be any non-uniform lattice of . Let
denote the number of subgroups of index at most in .
Then the limit exists and equals a constant which depends only on
the Lie type of and can be easily computed from its root system.Comment: 34 page
- …