BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Mima Stanojkovski (University of Trento) DTSTART:20240516T090000Z DTEND:20240516T095000Z DTSTAMP:20260416T184036Z UID:GiG2024/1 DESCRIPTION:Title: Studying p-groups via their Pfaffians: isomorphism testing and the PORC co njecture\nby Mima Stanojkovski (University of Trento) as part of Group s in Galway 2024\n\nLecture held in McMunn lecture theatre.\n\nAbstract\nG iven a field $K$\, to each alternating $n \\times n$ matrix of linear form s in $K[y_1\,\\dots \,y_d]$ one can associate a group scheme $\\mathrm{G}$ over $K$. In particular\, when $K$ is the field of rationals and $F$ is t he field of $p$ elements\, the $F$-points $\\mathrm{G}(F)$ of $\\mathrm{G} $ form a group of order $p^{n+d}$ and so\, as $p$ varies\, one obtains an infinite family of $p$-groups from $\\mathrm{G}$. In this talk\, I will pr esent joint work with Josh Maglione and Christopher Voll\, as well as ongo ing work with Eamonn O'Brien\, on the geometric study of automorphisms and isomorphism types of groups associated to small values of the parameters $n$ and $d$. I will also explain the implications of our work in connectio n to claims made around Higman's famous PORC conjecture.\n LOCATION:https://researchseminars.org/talk/GiG2024/1/ END:VEVENT BEGIN:VEVENT SUMMARY:Matteo Vannacci (University of the Basque Country) DTSTART:20240516T103000Z DTEND:20240516T112000Z DTSTAMP:20260416T184036Z UID:GiG2024/2 DESCRIPTION:Title: Profinite groups of finite probabilistic virtual rank\nby Matteo Vanna cci (University of the Basque Country) as part of Groups in Galway 2024\n\ nLecture held in McMunn lecture theatre.\n\nAbstract\nA profinite group $G $ carries naturally the structure of a probability space\, namely with res pect to its normalised Haar measure. We study the probability $Q(G\,k)$ th at $k$ Haar-random elements generate an open subgroup in the profinite gro up $G$. In particular\, in this talk I will introduce the probabilistic vi rtual rank $\\mathrm{pvr}(G)$ of $G$\; that is\, the smallest $k$ such tha t $Q(G\,k)=1$. We will discuss some key theorems and open problems about random generation in profinite groups\, with a view toward finite direct p roducts of hereditarily just infinite profinite groups. Classic examples o f the latter type of groups are semisimple algebraic groups over non-archi medean local fields. This is joint work with Benjamin Klopsch and Davide V eronelli.\n LOCATION:https://researchseminars.org/talk/GiG2024/2/ END:VEVENT BEGIN:VEVENT SUMMARY:Mireille Soergel (Max Planck Institute for Mathematics in the Scie nces) DTSTART:20240516T130000Z DTEND:20240516T135000Z DTSTAMP:20260416T184036Z UID:GiG2024/3 DESCRIPTION:Title: Dyer groups: Coxter groups\, right-angled Artin groups and more\nby Mi reille Soergel (Max Planck Institute for Mathematics in the Sciences) as p art of Groups in Galway 2024\n\nLecture held in McMunn lecture theatre.\n\ nAbstract\nDyer groups are a family encompassing both Coxeter groups and\n right-angled Artin groups. Each of these two classes of groups have\nnatur al piecewise Euclidean CAT(0) spaces associated to them: the\nDavis-Mousso ng complex for Coxeter groups and the Salvetti complex for\nright-angled A rtin groups. In this talk I will introduce Dyer groups\,\ngive some of the ir properties.\n LOCATION:https://researchseminars.org/talk/GiG2024/3/ END:VEVENT BEGIN:VEVENT SUMMARY:Andoni Zozaya (University of Ljubljana) DTSTART:20240516T143000Z DTEND:20240516T152000Z DTSTAMP:20260416T184036Z UID:GiG2024/4 DESCRIPTION:Title: Linearity of compact analytic groups over domains of characteristic zero\nby Andoni Zozaya (University of Ljubljana) as part of Groups in Galway 2024\n\nLecture held in McMunn lecture theatre.\n\nAbstract\nA $p$-adic a nalytic group is a topological group that is endowed with an analytic mani fold structure over $\\mathbb{Z}_p$\, the ring of $p$-adic integers. This definition can be extended by considering the manifold structure over more general pro-$p$ domains\, such as the power series rings $\\mathbb{Z}_p[[ t_1\, \\dots\, t_m]]$ or $\\mathbb{F}_p[[t_1\, \\dots\, t_m]]$ (where $\\m athbb{F}_p$ denotes the finite field of $p$ elements).\n\nLazard establish ed already in the 1960s that compact $p$-adic analytic groups are linear\, as they can be embedded as a closed subgroup within the group of invertib le matrices over $\\mathbb{Z}_p$. Nonetheless\, the question of the linear ity of analytic groups over more general domains remains unsolved.\n\nIn t his talk\, we shed some light to this question by proving that when the co efficient ring is of characteristic zero\, every compact analytic group is linear. We will provide background on the problem and outline the strateg y of our argument.\n\nJoint with M. Casals-Ruiz.\n LOCATION:https://researchseminars.org/talk/GiG2024/4/ END:VEVENT BEGIN:VEVENT SUMMARY:Itay Glazer (University of Oxford) DTSTART:20240516T153000Z DTEND:20240516T162000Z DTSTAMP:20260416T184036Z UID:GiG2024/5 DESCRIPTION:Title: Fourier and Small ball estimates for word maps on unitary groups\nby I tay Glazer (University of Oxford) as part of Groups in Galway 2024\n\nLect ure held in McMunn lecture theatre.\n\nAbstract\nLet $w(x\,y)$ be a word i n a free group. For any group $G$\, $w$ induces a word map $w:G^2 \\to G$. For example\, the commutator word $w=xyx^{-1}y^{-1}$ induces the commutat or map. If $G$ is finite\, one can ask what is the probability that $w(g\, h)$ is equal to the identity element $e$\, for a pair $(g\,h)$ of elements in $G$\, chosen independently at random. \nIn the setting of finite simpl e groups\, Larsen and Shalev showed there exists $\\epsilon(w)>0$ (dependi ng only on $w$)\, such that the probability that $w(g\,h)=e$ is smaller th an $|G|^{-\\epsilon(w)}$\, whenever $G$ is large enough (depending on $w$) . \nIn this talk\, I will discuss analogous questions for compact groups\, with a focus on the family of unitary groups\; For example\, given a word $w$\, and given two independent Haar-random $n$ by $n$ unitary matrices $ A$ and $B$\, what is the probability that $w(A\,B)$ is contained in a smal l ball around the identity matrix?\n\nBased on a joint work with Nir Avni and Michael Larsen.\n LOCATION:https://researchseminars.org/talk/GiG2024/5/ END:VEVENT BEGIN:VEVENT SUMMARY:Waltraud Lederle (Dresden University of Technology) DTSTART:20240517T083000Z DTEND:20240517T092000Z DTSTAMP:20260416T184036Z UID:GiG2024/6 DESCRIPTION:Title: Boomerang subgroups\nby Waltraud Lederle (Dresden University of Techno logy) as part of Groups in Galway 2024\n\nLecture held in McMunn lecture t heatre.\n\nAbstract\nGiven a locally compact group\, its set of closed sub groups can be endowed with a compact\, Hausdorff topology. With this topol ogy\, it is called the Chabauty space of the group. Every group acts on it s Chabauty space via conjugation. This action has connections to rigidity theory\, Margulis' normal subgroup theorem and measure preserving actions of the group via so-called Invariant Random Subgroups (IRS).\nI will give a gentle introduction into Chabauty spaces and IRS and state a few classic al results. I will define boomerang subgroups and explain how special case s of the classical results can be proven via them.\nBased on joint work wi th Yair Glasner.\n LOCATION:https://researchseminars.org/talk/GiG2024/6/ END:VEVENT BEGIN:VEVENT SUMMARY:Anna Giordano Bruno (University of Udine) DTSTART:20240517T100000Z DTEND:20240517T105000Z DTSTAMP:20260416T184036Z UID:GiG2024/7 DESCRIPTION:Title: A brief history and recent advances in the theory of characterized subgrou ps of the circle group\nby Anna Giordano Bruno (University of Udine) a s part of Groups in Galway 2024\n\nLecture held in McMunn lecture theatre. \n\nAbstract\nA subgroup $H$ of the circle group $\\mathbb T$ is said to b e characterized by a sequence of integers $\\mathbf u = (u_n)_{n\ \in\\mathbb N}$ if $H=\\{x\\in\\mathbb T: u_nx\\to 0\\}$. The first part o f the talk discusses characterized subgroups of $\\mathbb T$ and their rel evance in several areas of Mathematics where the behavior of the sequence $(u_nx)_{n\\in\\mathbb N}$ as above is studied\, as Topological Algebra (t opologically torsion elements and characterized subgroups)\, Harmonic Anal ysis (sets of convergence of trigonometric series\, thin sets) and Number Theory (uniform distribution of sequences).\n\nRecently\, generalizations of the notion of characterized subgroup of $\\mathbb T$ were introduced\, based on weaker notions of convergence\, starting from statistical converg ence and ending with $\\mathcal I$-convergence for an ideal $\\mathcal I$ of $\\mathbb N$\, due to Cartan. A sequence $(y_n)_{n\\in\\mathbb N}$ in $ \\mathbb T$ is said to $\\mathcal I$-converge to a point $y\\in \ \mathbb T$\, denoted by $y_n\\overset{\\mathcal I}\\to y$\, if $\\{n\\in\\ mathbb N: y_n \\not \\in U\\}\\in \\mathcal I$ for every neighborhood $U$ of $y$ in $\\mathbb T$. A subgroup $H$ of the circle group $\\mathbb T$ is said to be $\\mathcal I$-characterized with respect to $\\mathca l I$ by a sequence of integers $\\mathbf u = (u_n)_{n\\in\\mathbb N}$ if $ H=\\{x\\in\\mathbb T: u_nx\\overset{\\mathcal I}\\to 0\\}$. The second par t of the presentation proposes an overview on the results obtained on thes e new kind of characterized subgroups\, with special emphasis on $\\mathca l I$-characterized subgroups of $\\mathbb T$.\n\nBased on a joint work wit h D. Dikranjan\, R. Di Santo and H. Weber.\n LOCATION:https://researchseminars.org/talk/GiG2024/7/ END:VEVENT BEGIN:VEVENT SUMMARY:Alex Evetts (University of Manchester) DTSTART:20240517T110000Z DTEND:20240517T115000Z DTSTAMP:20260416T184036Z UID:GiG2024/8 DESCRIPTION:Title: Twisted conjugacy growth of virtually nilpotent groups\nby Alex Evetts (University of Manchester) as part of Groups in Galway 2024\n\nLecture he ld in McMunn lecture theatre.\n\nAbstract\nThe conjugacy growth function o f a finitely generated group is a variation of the standard growth functio n\, counting the number of conjugacy classes intersecting the $n$-ball in the Cayley graph. The asymptotic behaviour is not a commensurability invar iant in general\, but the conjugacy growth of finite extensions can be und erstood via the twisted conjugacy growth function\, counting automorphism- twisted conjugacy classes. I will discuss what is known about the asymptot ic and formal power series behaviour of (twisted) conjugacy growth\, in pa rticular some relatively recent results for certain groups of polynomial g rowth (i.e. virtually nilpotent groups).\n LOCATION:https://researchseminars.org/talk/GiG2024/8/ END:VEVENT END:VCALENDAR