Dr Colin Reid

We obtain some global features of totally disconnected locally compact (t.d.l.c.) groups G that are locally isomorphic to a just infinite profinite group, building on an earlier result of Barnea¿Ershov¿Weigel and also using tools developed by P.-E. Caprace, G. Willis and the author for studying local structure in t.d.l.c. groups. The approach uses the following property of just infinite profinite groups, essentially due to Wilson: given a locally normal subgroup K of G, then there is an open subgroup of K that is a direct factor of an open subgroup of G. This is a local property of t.d.l.c. groups and we obtain a characterization of the local isomorphism types of t.d.l.c. groups that have it.

We show that, given a compact minimal system and an element h of the topological full group of g, the infinite orbits of h admit a locally constant orientation with respect to the orbits of g. We use this to obtain a clopen partition of into minimal and periodic parts, where G is any virtually polycyclic subgroup of. We also use the orientation of orbits to give a refinement of the index map and to describe the role in of the submonoid generated by the induced transformations of g. Finally, we consider the problem, given a homeomorphism h of the Cantor space X, of determining whether or not there exists a minimal homeomorphism g of X such that.

In finite group theory, chief factors play an important and well-understood role in the structure theory. We here develop a theory of chief factors for Polish groups. In the development of this theory, we prove a version of the Schreier refinement theorem. We also prove a trichotomy for the structure of topologically characteristically simple Polish groups. The development of the theory of chief factors requires two independently interesting lines of study. First we consider injective, continuous homomorphisms with dense normal image. We show such maps admit a canonical factorisation via a semidirect product, and as a consequence, these maps preserve topological simplicity up to abelian error. We then define two generalisations of direct products and use these to isolate a notion of semisimplicity for Polish groups.

We construct uncountably many non-locally isomorphic examples of topologically simple nondiscrete totally disconnected locally compact groups. The new examples differ from known examples of such groups in that they have trivial quasi-centre, but also have infinite abelian locally normal subgroups. The examples are constructed as almost upper-triangular matrices modulo scalar matrices over finite fields, where 'almost upper-triangular' is defined with respect to one of an uncountable family of preorders generalising the orders (Z, =) and (N, =).

Let G be a totally disconnected locally compact (t.d.l.c.) group and let H be an equicontinuously (for example, compactly) generated group of automorphisms of G. We show that every distal action of H on a coset space of G is a SIN action, with the small invariant neighborhoods arising from open H-invariant subgroups. We obtain a number of consequences for the structure of the collection of open subgroups of a t.d.l.c. group. For example, it follows that for every compactly generated subgroup K of G, there is a compactly generated open subgroup E of G such that K = E and such that every open subgroup of G containing a finite index subgroup of K contains a finite index subgroup of E. We also show that for a large class of closed subgroups L of G (including for instance all closed subgroups L such that L is an intersection of subnormal subgroups of open subgroups), every compactly generated open subgroup of L can be realized as L ? O for an open subgroup of G.

We give criteria on an inverse system of finite groups that ensure that the limit is just infinite or hereditarily just infinite. More significantly, these criteria are 'universal' in that all (hereditarily) just infinite profinite groups arise as limits of the specified form. Inspired by a recent paper of Wilson, we give special consideration to (hereditarily) just infinite profinite groups that are not virtually pro-p. © 2011 London Mathematical Society.

Let G be a finite non-nilpotent group such that every Sylow subgroup of G is generated by at most d elements, and such that p is the largest prime dividing {pipe}G{pipe}. We show that G has a non-nilpotent image G/N, such that N is characteristic and of index bounded by a function of d and p. This result will be used to prove that G has a nilpotent normal subgroup of index bounded in terms of d and p. © 2011 The Author(s).

A residually finite (profinite) group G is just infinite if every non-trivial (closed) normal subgroup of G is of finite index. This paper considers the problem of determining which (closed) subgroups of finite index of a just infinite group are themselves just infinite. If G is just infinite and not virtually abelian, we show that G is hereditarily just infinite if and only if all maximal (closed) subgroups of finite index are just infinite. This result will be used to show that a finitely generated pro-p group G is just infinite if and only if G has no non-trivial finite normal subgroups and F(G) has a just infinite open subgroup. © Elsevier Inc.

A profinite group G is just infinite if every closed normal subgroup of G is of finite index. We prove that an infinite profinite group is just infinite if and only if, for every open subgroup H of G, there are only finitely many open normal subgroups of G not contained in H. This extends a result recently established by Barnea, Gavioli, Jaikin-Zapirain, Monti and Scoppola (2009) in [1], who proved the same characterisation in the case of pro-p groups. We also use this result to establish a number of features of the general structure of profinite groups with regard to the just infinite property. © Elsevier Inc.