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### CARMA-Sponsored Seminar Series: Colloquia, Seminars and More.

[Note: events are listed by descending date.]
 ZERO-DIMENSIONAL SYMMETRY SEMINAR Speaker: Mr Davide Spriano, ETH Zürich Title: Convexity and generalization of hyperbolicity Location: Room MC102, McMullin (Callaghan Campus) The University of Newcastle Time and Date: 2:00 pm, Tue, 11th Dec 2018 Abstract: Almost by definition, the main tool and goal of Geometric Group Theory is to find and exploit correspondences between geometric and algebraic features of groups. Following this philosophy, I will focus on the question: what does it mean for a sub(space/group) to "sit nicely" inside a bigger (space/group)? Focusing on groups, for a subgroup H of a group G, possible answers for the above question are when the subgroup H is: quasi-isometrically embedded, undistorted, normal/malnormal, finitely generated, geometrically separated... Many of the above are equivalent when H is a quasiconvex subgroup of a hyperbolic group G, providing very successful correspondences between geometric and algebraic properties of subgroups. The goal of this talk is to review quasiconvexity in hyperbolic spaces and try to generalize several of those features in a broader setting, namely the class of hierarchically hyperbolic groups (HHG). This is a joint work with Hung C. Tran and Jacob Russell. [Permanent event link] ZERO-DIMENSIONAL SYMMETRY SEMINAR Speaker: Alejandra Garrido, The University of Newcastle Title: Hausdorff dimension and normal subgroups of free-like pro-p groups Location: Room LG 17, McMullin (Callaghan Campus) The University of Newcastle Time and Date: 2:00 pm, Tue, 27th Nov 2018 Abstract: Hausdorff dimension has become a standard tool to measure the "size" of fractals in real space. However, it can be defined on any metric space and therefore can be used to measure the "size" of subgroups of, say, pro-p groups (with respect to a chosen metric). This line of investigation was started 20 years ago by Barnea and Shalev, who showed that p-adic analytic groups do not have any "fractal" subgroups, and asked whether this characterises them among finitely generated pro-p groups. I will explain what all of this means and report on joint work with Oihana Garaialde and Benjamin Klopsch in which, while trying to solve this problem, we ended up showing an analogue of a theorem of Schreier in the context of pro-p groups of positive rank gradient: any finitely generated infinite normal subgroup of a pro-p group of positive rank gradient is of finite index. I will also explain what "positive rank gradient" means, and why pro-p groups with such a property are "free-like". [Permanent event link] ZERO-DIMENSIONAL SYMMETRY SEMINAR Speaker: Dr Thibaut Dumont, University of Jyväskylä Title: Cocycles on trees and piecewise translation action on locally compact groups Location: Room MC110, McMullin Building (Callaghan Campus) The University of Newcastle Time and Date: 2:00 pm, Tue, 20th Nov 2018 Abstract: In the first part of this seminar, I will present some geometric cocycles associated to trees and ways to compute their norms. Similar construction exists for Euclidean buildings but no satisfactory estimates of the norm is currently known. In the second part, I will discuss some ongoing research with Thibaut Pillon on actions the infinite cyclic group by piecewise translations on locally compact group. Piecewise translation actions have been well studied for finitely generated groups, e.g. by Whyte, and provide positive answers to the von-Neumann-Day problem or the Burnside problem. The generalization to LC-groups was introduced by Schneider. The topic seems to have interesting implications for tdlc-groups [Permanent event link] ZERO-DIMENSIONAL SYMMETRY SEMINAR Speaker: Anne Thomas, The University of Sydney Title: Divergence in right-angled Coxeter groups Location: Room MC102, McMullin (Callaghan Campus) The University of Newcastle Time and Date: 2:00 pm, Mon, 12th Nov 2018 Abstract: The divergence of a pair of geodesics in a metric space measures how fast they spread apart. For example, in Euclidean space all pairs of geodesics diverge linearly, while in hyperbolic space all pairs of geodesics diverge exponentially. In the 1980s Gromov proved that in symmetric spaces of non-compact type, the only possible divergence rates are linear or exponential, and he asked whether the same dichotomy holds in CAT(0) spaces. Soon afterwards, Gersten used these ideas to define a quasi-isometry invariant, also called divergence, which measures the "worst" rate of divergence. Gersten and others have since found many examples of finitely generated groups with quadratic divergence. We study divergence in right-angled Coxeter groups with triangle-free defining graphs. Using the structure of certain flats in the associated Davis complex, which is a CAT(0) square complex, we characterise such groups with linear and quadratic divergence, and construct examples of right-angled Coxeter groups with divergence polynomial of arbitrary degree. This is joint work with Pallavi Dani (Louisiana State University). [Permanent event link] ZERO-DIMENSIONAL SYMMETRY SEMINAR Speaker: Alejandra Garrido, The University of Newcastle Title: Maximal subgroups of some groups of intermediate growth Location: Room LG 17, McMullin (Callaghan Campus) The University of Newcastle Time and Date: 2:00 pm, Tue, 16th Oct 2018 Abstract: Given a group one of the most natural things one can study about it is its subgroup lattice, and the maximal subgroups take a prominent role. If the group is infinite, one can ask whether all maximal subgroups have finite index or whether there are some (and how many) of infinite index. After telling some historical developments on this question, I will motivate the study of maximal subgroups of groups of intermediate growth and report on joint work with Dominik Francoeur where we give a complete description of all maximal subgroups of some "siblings" of Grigorchuk's group. [Permanent event link] ZERO-DIMENSIONAL SYMMETRY SEMINAR Speaker: , CARMA, The University of Newcastle Title: Algebraic theory of self-similar groups Location: Room LG 17, McMullin (Callaghan Campus) The University of Newcastle Time and Date: 2:00 pm, Tue, 9th Oct 2018 Abstract: I will describe the relationship between self-similar groups, permutational bimodules and virtual group endomorphisms. Based on chapter 2 of Nekrashevych’s book. [Permanent event link] ZERO-DIMENSIONAL SYMMETRY SEMINAR The Group Co-Word Problem Speaker: Mr Alex Bishop, University of Technology Sydney Title: The Group Co-Word Problem Location: Room LG 17, McMullin (Callaghan Campus) The University of Newcastle Time and Date: 2:00 pm, Tue, 2nd Oct 2018 Abstract: In this talk, we will introduce a class of tree automorphism groups known as bounded automata. From this definition, we will see that many of the interesting examples of self-similar groups in the literature are members of this class. A problem in group theory is classifying groups based on the difficulty of solving their co-word problems, that is, classifying them by the computational difficulty to decide if a word is not equivalent to the identity. Some well-known results in this study are that a group has a co-word problem given by a regular language if and only if it is finite, a deterministic context-free language if and only if it is virtually free, and a deterministic one-counter machine if and only if it is virtually cyclic. Each of these language classes corresponds to a natural and well-studied model of computation. We will show that the class of bounded automata groups has a co-word problem given by an ET0L language – a class of formal language which has recently gained popularity in areas of group theory. This strengthens a recent result of Holt and Röver (who showed this result for a less restrictive class of language) and extends a result of Ciobanu-Elder-Ferov (who proved this result for the first Grigorchuk group). [Permanent event link] ZERO-DIMENSIONAL SYMMETRY SEMINAR Speaker: Mr Timothy Bywaters, The University of Sydney Title: Spaces at infinity for hyperbolic totally disconnected locally compact groups Location: Room LG 17, McMullin (Callaghan Campus) The University of Newcastle Time and Date: 2:00 pm, Tue, 25th Sep 2018 Abstract: Every compactly generated t.d.l.c. group acts vertex transitively on a locally finite graph with compact open vertex stabilisers. Such a graph is called a rough Cayley graph and, up to quasi-isometry, is an invariant for the group. This allows us to define Gromov hyperbolic t.d.l.c. groups and their Gromov boundary in a way analogous to the finitely generated case. The space of directions of a t.d.l.c. group is a metric space 'at infinity' obtained by analysing the action of the group on the set of compact open subgroups. It is particularly useful for detecting flat subgroups, think subgroups that look like $\mathbb{Z}^n$. In my talk, I will introduce these two concepts of boundary and give some new results which relate them. Time permitting, I may also give details about the proofs. [Permanent event link] 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 event link] ZERO-DIMENSIONAL SYMMETRY SEMINAR Speaker: Dr Stephan Tornier, The University of Newcastle Title: An introduction to self-similar groups Location: Room MC102, McMullin (Callaghan Campus) The University of Newcastle Time and Date: 2:00 pm, Mon, 3rd Sep 2018 Abstract: We introduce the notion of self-similarity for groups acting on regular rooted trees as well as their description using automata and wreath iteration. Following the definition of Grigorchuk's group we show that it is an infinite, finitely generated 2-group. The proof illustrates the use of self-similarity. [Permanent event link] ZERO-DIMENSIONAL SYMMETRY SEMINAR Speaker: ARC Laureate Fellow George Willis, CARMA, The University of Newcastle Title: The tree representation theorem and automorphism groups of rooted trees Location: Room MC102, McMullin (Callaghan Campus) The University of Newcastle Time and Date: 2:00 pm, Mon, 27th Aug 2018 Abstract: (joint work with R. Grigorchuk ad D. Horadam) The tree representation theorem represents a certain group associated with the scale of an automorphism of a t.d.l.c. group as acting by symmetries of a regular (unrooted) tree. It shows that groups acting on regular trees are a fundamental part of the theory of t.d.l.c. groups. There is also an extensive theory of self-similar and self-replicating groups of symmetries of rooted trees which has developed from the discovery (or creation) of examples such as the Grigorchuk groups. It will be seen in this talk that these two branches of research are studying essentially the same groups. [Permanent event link] ZERO-DIMENSIONAL SYMMETRY SEMINAR Speaker: ARC Laureate Fellow George Willis, CARMA, The University of Newcastle Title: Locally pro-p contraction groups are nilpotent Location: Room MC102, McMullin (Callaghan Campus) The University of Newcastle Dates: Mon, 20th Aug 2018 - Mon, 20th Aug 2018 Abstract: See here for an abstract. [Permanent event link] ZERO-DIMENSIONAL SYMMETRY SEMINAR Speaker: Dr Michal Ferov, CARMA, The University of Newcastle Title: Separating cyclic subgroups in graph products of groups Location: Room MC102, McMullin (Callaghan Campus) The University of Newcastle Dates: Mon, 13th Aug 2018 - Mon, 13th Aug 2018 Abstract: (joint work with Federico Berlai) A natural way to study infinite groups is via looking at their finite quotients. A subset S of a group G is then said to be (finitely) separable in G if we can recognise it in some finite quotient of G, meaning that for every g outside of S there is a finite quotient of G such that the image of g under the canonical projection does not belong to the image of S. We can then describe classes of groups by specifying which types of subsets do we require to be separable: residually finite groups have separable singletons, conjugacy separable groups have separable conjugacy classes of elements, cyclic subgroup separable groups have separable cyclic subgroups and so on... We could also restrict our attention only to some class of quotients, such as finite p-groups, solvable, alternating... Properties of this type are called separability properties. In case when the class of admissible quotients has reasonable closure properties we can use topological methods. We prove that the property of being cyclic subgroup separable, that is having all cyclic subgroups closed in the profinite topology, is preserved under forming graph products. Furthermore, we develop the tools to study the analogous question in the pro-p case. For a wide class of groups we show that the relevant cyclic subgroups - which are called p-isolated - are closed in the pro-p topology of the graph product. In particular, we show that every p-isolated cyclic subgroup of a right-angled Artin group is closed in the pro-p topology and, consequently, we show that maximal cyclic subgroups of a right-angled Artin group are p-separable for every p. [Permanent event link]