Margulis’s normal subgroup theorem A short introduction

The normal subgroup theorem of Margulis expresses that many lattices in semi-simple Lie groups are simple up to finite error. In this talk, we give a short introduction to the normal subgroup theorem , including a brief review of the relevant notions about lattices, amenability, and property (T), as well as a short overview of the proof of the normal subgroup theorem. 1 Margulis's normal subgroup theorem The normal subgroup theorem of Margulis expresses that many lattices in semi-simple Lie group are simple groups up to finite error: Theorem 1.1 (Normal subgroup theorem). Let G be a connected semi-simple Lie group with finite centre with rk R (G) ≥ 2, and let Γ ⊂ G be an irreducible lattice. If N ⊂ Γ is a normal subgroup of Γ, then either N lies in the centre of G (and hence Γ is finite) or the quotient Γ/N is finite. Remark 1.2. In fact, Margulis proved the normal subgroup theorem for a more general class of lattices – namely, lattices in certain ambient groups that are " Lie groups " over local fields [9, Chapter IV]. After explaining the occuring terminology in detail in Section 2, we discuss the statement and the consequences of the normal subgroup theorem in Section 3. Sections 4 and 5 give a short introduction into amenability and property (T), which lie at the heart of the proof of the normal subgroup theorem. Finally, Section 6 is devoted to an overview of the proof of the normal subgroup theorem. For a more extensive treatment of the normal subgroup theorem and related topics, we refer the reader to the books by