MathOverflow is a question and answer site for professional mathematicians. It only takes a minute to sign up. It is known, from the works of G. Margulis, etc. Do they admit explicit constructions like those coming from global fields? All such lattices in the higher rank case are S-arithmetic. When all but one of the factors is compact, the projection onto the non-compact factor that you can take to be a p-adic field gives you a lattice there.

## Discrete Subgroups of Lie Groups | Madabusi S. Raghunathan | Springer

The projection onto the first component is dense, and onto the second component is a lattice there. For the precise statement of the S- arithmeticity theorem, the best place to look is Margulis's book "Discrete Subgroups of Semisimple Lie Group". Sign up to join this community.

Auxiliary Assertions. The Multiplicative Ergodic Theorem. Definition and Fundamental Properties of Characteristic Maps. Effective Pairs. Essential Pairs. Discrete Subgroups and Boundary Theory. Proximal G-Spaces and Boundaries. Projective G-Spaces. Equivariant Measurable Maps to Algebraic Varieties. Cocycles on G-Spaces.

Finite-Dimensional Invariant Subspaces. Rigidity of Ergodic Actions of Semisimple Groups. Statement of the Arithmeticity Theorems. Proof of the Arithmeticity Theorems. Finite Generation of Lattices. Consequences of the Arithmeticity Theorems I. Consequences of the Arithmeticity Theorems II.

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## Discrete Subgroups of Lie Groups

Table of contents 1. Statement of Main Results. Synopsis of the Chapters. Remarks on the Structure of the Book, References and Notation. Notation, Terminology and Some Basic Facts. Algebraic Groups Over Arbitrary Fields. Algebraic Groups Over Local Fields. Arithmetic Groups.

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