Locally normal subgroups of totally disconnected groups. Part I: General theory

Let G be a totally disconnected, locally compact group. A locally normal subgroup of G is a compact subgroup whose normaliser is open. We begin an investigation of the structure of the family of locally normal subgroups of G. Modulo commensurability, this family forms a modular lattice LN(G), called the structure lattice of G. We show that G admits a canonical maximal quotient H for which the quasi-centre and the abelian locally normal subgroups are trivial. In this situation LN(H) has a canonical subset called the centraliser lattice, forming a Boolean algebra whose elements correspond to centralisers of locally normal subgroups. If H acts faithfully on its centraliser lattice, we show that the topology of H is determined by its algebraic structure (and thus invariant by every abstract group automomorphism), and also that the action on the Stone space of the centraliser lattice is universal for a class of actions on profinite spaces.