ZERO-DIMENSIONAL SYMMETRY SEMINAR Speaker: Dr Colin Reid, CARMA, The University of Newcastle Title: Endomorphisms of profinite groups Location: Room MC102, McMullin (Callaghan Campus) The University of Newcastle Time and Date: 2:00 pm, Mon, 10th Sep 2018 Abstract: Given a profinite group $G$, we can consider the semigroup $\mathrm{End}(G)$ of continuous homomorphisms from $G$ to itself. In general $\lambda \in\mathrm{End}(G)$ can be injective but not surjective, or vice versa: consider for instance the case when $G$ is the group $F_p[[t]$ of formal power series over a finite field, $n$ is an integer, and $\lambda_n$ is the continuous endomorphism that sends $t^k$ to $t^{k+n}$ if $k+n \ge 0$ and $0$ otherwise. However, when $G$ has only finitely many open subgroups of each index (for instance, if $G$ is finitely generated), the structure of endomorphisms is much more restricted: given $\lambda \in\mathrm{End}(G)$, then $G$ can be written as a semidirect product $N \rtimes H$ of closed subgroups, where $\lambda$ acts as an automorphism on $H$ and a contracting endomorphism on $N$. When $\lambda$ is open and injective, the structure of $N$ can be restricted further using results of Glöckner and Willis (including the very recent progress that George told us about a few weeks ago). This puts some restrictions on the profinite groups that can appear as a '$V_+$' group for an automorphism of a t.d.l.c. group. [Permanent link] CARMA COLLOQUIUM Speaker: Dr Colin Reid, CARMA, The University of Newcastle Title: Totally disconnected, locally compact groups Location: Room V205, Mathematics Building (Callaghan Campus) The University of Newcastle Time and Date: 4:00 pm, Tue, 28th Mar 2017 Abstract: Totally disconnected, locally compact (t.d.l.c.) groups are a large class of topological groups that arise from a few different sources, for instance as automorphism groups of a range of algebraic and combinatorial structures, or from the study of isomorphisms between finite index subgroups of a given group. A general theory has begun to emerge in recent years, based on the interaction between small-scale and large-scale structure in t.d.l.c. groups. I will give a survey of some ways in which these groups arise and some of the tools that have been developed for understanding them. [Permanent link] CARMA GROUP THEORY RHD MEETING Speaker: Dr Colin Reid, CARMA, The University of Newcastle Title: Proof of the p-localisation theorem Location: Room V205, Mathematics Building (Callaghan Campus) The University of Newcastle Time and Date: 11:00 am, Thu, 13th Oct 2016 [Permanent link] CARMA GROUP THEORY RHD MEETING Speaker: Dr Colin Reid, CARMA, The University of Newcastle Title: Introduction to p-localisations Location: Room V205, Mathematics Building (Callaghan Campus) The University of Newcastle Time and Date: 11:00 am, Thu, 1st Sep 2016 [Permanent link] GROUPS AND GEOMETRY RHD MEETING Speaker: Dr Colin Reid, CARMA, The University of Newcastle Title: Groups and Geometry RHD Meeting Location: Room V205, Mathematics Building (Callaghan Campus) The University of Newcastle Time and Date: 2:00 pm, Wed, 30th Jul 2014 Abstract: Colin Reid will present some thoughts on limits of contraction groups. [Permanent link] CARMA COLLOQUIUM Speaker: Dr Colin Reid, CARMA, The University of Newcastle Title: Locally normal subgroups of totally disconnected groups Location: Room V129, Mathematics Building (Callaghan Campus) The University of Newcastle Time and Date: 4:00 pm, Thu, 1st Nov 2012 Abstract: I will give an extended version of my talk at the AustMS meeting about some ongoing work with Pierre-Emmanuel Caprace and George Willis. Given a locally compact topological group G, the connected component of the identity is a closed normal subgroup G_0 and the quotient group is totally disconnected. Connected locally compact groups can be approximated by Lie groups, and as such are relatively well-understood. By contrast, totally disconnected locally compact (t.d.l.c.) groups are a more difficult class of objects to understand. Unlike in the connected case, it is probably hopeless to classify the simple t.d.l.c. groups, because this would include for instance all simple groups (equipped with the discrete topology). Even classifying the finitely generated simple groups is widely regarded as impossible. However, we can prove some general results about broad classes of (topologically) simple t.d.l.c. groups that have a compact generating set. Given a non-discrete t.d.l.c. group, there is always an open compact subgroup. Compact totally disconnected groups are residually finite, so have many normal subgroups. Our approach is to analyse a t.d.l.c. group G (which may itself be simple) via normal subgroups of open compact subgroups. From these we obtain lattices and Cantor sets on which G acts, and we can use properties of these actions to demonstrate properties of G. For instance, we have made some progress on the question of whether a compactly generated topologically simple t.d.l.c. group is abstractly simple, and found some necessary conditions for G to be amenable. [Permanent link]