Giles Gardam
I am a postdoc in geometric group theory working in the Topology Group at the WWU Münster (job title: "akademischer Rat a.Z." ~ research associate). I did my graduate studies under Martin Bridson's supervision in Oxford, defending my DPhil thesis on 1 September 2017. Afterwards I was a postdoc for a year at the Technion before participating in the Hausdorff Trimester Program Logic and Algorithms in Group Theory in Bonn (September to December 2018). I did my undergraduate studies at the University of Sydney and wrote my honours thesis under the supervision of Anne Thomas.
 Email: <first initial><last name>@unimuenster.de
 Post: Mathematisches Institut, Einsteinstr. 62, 48149 Münster, Germany
 Office: 513
 Pronouns: he/him
 Pronunciation: dʒaɪlz starts like giant and rhymes with miles
Research interests
I'm interested in obstructions to negative and nonpositive curvature (for instance, in what settings are subgroups like BaumslagSolitar groups the only obstruction?), group rings, onerelator groups, group boundaries, finiteness properties, profinite rigidity, minimal presentations (e.g. deficiency), group laws, algorithms and compexity in group theory. My interest in computational group theory is not purely theoretical: I code as appropriate.
Papers
My papers are all on my arXiv page.
New preprints

A counterexample to the unit conjecture for group rings, arXiv:2102.11818 pdf
announcement talk at Symmetry in Newcastle (Australia) on youtube, slides from NYGT talkThe unit conjecture, commonly attributed to Kaplansky, predicts that if \(K\) is a field and \(G\) is a torsionfree group then the only units of the group ring \(K[G]\) are the trivial units, that is, the nonzero scalar multiples of group elements. We give a concrete counterexample to this conjecture; the group is virtually abelian and the field is order two.

JSJ decompositions and polytopes for twogenerator onerelator groups, with Dawid Kielak and Alan Logan, arXiv:2101.02193
We provide a direct connection between the \(\mathcal{Z}_{\max}\) (or essential) JSJ decomposition and the Friedl–Tillmann polytope of a hyperbolic twogenerator onerelator group with abelianisation of rank \(2\). We deduce various structural and algorithmic properties, like the existence of a quadratictime algorithm computing the \(\mathcal{Z}_{\max}\)JSJ decomposition of such groups.
Published or accepted for publication

Finite groups of arbitrary deficiency, arXiv:1705.02040 journal version
Bull. London Math. Soc. 49 (2017), 1100–1104The deficiency of a group is the maximum over all presentations for that group of the number of generators minus the number of relators. Every finite group has nonpositive deficiency. We show that every nonpositive integer is the deficiency of a finite group – in fact, of a finite \(p\)group for every prime \(p\). This completes Kotschick's classification of the integers which are deficiencies of fundamental groups of compact Kähler manifolds.

The geometry of onerelator groups satisfying a polynomial isoperimetric inequality with Daniel Woodhouse, arXiv:1711.08755 journal version
Proc. Amer. Math. Soc. 147 (2019), 125–129For every pair of positive integers \(p > q\) we construct a onerelator group \(R_{p,q}\) whose Dehn function is \(\simeq n^\alpha\) where \(\alpha = 2 \log_2(2p / q)\). The group \(R_{p,q}\) has no subgroup isomorphic to a BaumslagSolitar group \(BS(m,n)\) with \(m \neq \pm n\), but is not automatic, not CAT(0), and cannot act freely on a CAT(0) cube complex. This answers a longstanding question on the automaticity of onerelator groups and gives counterexamples to a conjecture of Wise.

Detecting laws in power subgroups, arXiv:1705.09348 journal version
Comm. Algebra 47 (2019), 1699–1707(Note: arXiv version will be updated after embargo period from journal.)A group law is said to be detectable in power subgroups if, for all coprime \(m\) and \(n\), a group \(G\) satisfies the law if and only if the power subgroups \(G^m\) and \(G^n\) both satisfy the law. We prove that for all positive integers \(c\), nilpotency of class at most \(c\) is detectable in power subgroups, as is the \(k\)Engel law for \(k\) at most 4. In contrast, detectability in power subgroups fails for solvability of given derived length: we construct a finite group \(W\) such that \(W^2\) and \(W^3\) are metabelian but \(W\) has derived length \(3\). We analyse the complexity of the detectability of commutativity in power subgroups, in terms of finite presentations that encode a proof of the result.

Profinite rigidity in the SnapPea census, arXiv:1805.02697 journal version
to appear in Exp. Math.(Note: arXiv version will be updated after embargo period from journal.)A wellknown question asks whether any two nonisometric finite volume hyperbolic 3manifolds are distinguished from each other by the finite quotients of their fundamental groups. At present, this has been proved only when one of the manifolds is a oncepunctured torus bundle over the circle. We give substantial computational evidence in support of a positive answer, by showing that no two manifolds in the SnapPea census of 72 942 finite volume hyperbolic 3manifolds have the same finite quotients.

Minimal sizeable graphs, arXiv:1808.09505 appendix to the paper Hyperbolic Groups with Finitely Presented Subgroups not of Type F_{3} of R. Kropholler
to appear in Geom. Dedicata
Preprints

Cannon–Thurston maps for CAT(0) groups with isolated flats, with Benjamin Beeker, Matthew Cordes, Radhika Gupta, and Emily Stark, arXiv:1810.13285 (submitted)
Mahan Mitra (Mj) proved Cannon–Thurston maps exist for normal hyperbolic subgroups of a hyperbolic group. We prove that Cannon–Thurston maps do not exist for infinite torsionfree normal hyperbolic subgroups of a nonhyperbolic CAT(0) group with isolated flats with respect to the visual boundaries. We also show Cannon–Thurston maps do not exist for infinite torsionfree infiniteindex normal CAT(0) subgroups with isolated flats in a nonhyperbolic CAT(0) group with isolated flats. We determine the isomorphism types of the normal subgroups in the above two settings.
 Alternative proof of ballrestricted version, arXiv:1908.00830 appendix to the paper Two generalisations of Leighton's Theorem of S. Shepherd, joint with Daniel Woodhouse (submitted)
Upcoming talks
 ...
Recent talks
 Leighton's Theorem, BielefeldMünster Seminar on Groups, Geometry and Topology, 17 January 2020
 Onerelator groups and the Haagerup property, University of Vienna Geometry and Analysis on Groups Research Seminar, 14 January 2020
 The isomorphism problem for Coxeter groups, Flags, Galleries and Reflection Groups, University of Sydney, 8 August 2019
 Boundaries of hyperbolic and CAT(0) groups and CannonThurston maps, Bielefeld Oberseminar Groups and Geometry, 19 June 2019
 Boundaries of hyperbolic and CAT(0) groups and CannonThurston maps, Bonn Oberseminar Topologie, 16 April 2019
 Boundaries of hyperbolic and CAT(0) groups and CannonThurston maps, Münster Oberseminar Topologie, 28 January 2019
 Exotic onerelator groups, EPFL Ergodic and Geometric Group Theory Seminar, 17 May 2018
 Exotic onerelator groups, Neuchâtel Séminaire Groupes et Analyse, 15 May 2018
 Exotic onerelator groups, Regensburg SFBSeminar, 10 April 2018.
 Dehn functions of onerelator groups, Oxford Junior Topology and Geometry Seminar, 28 February 2018.
 Finding and not finding unusual onerelator groups, Winter OneRelator Workshop, 11 January 2018.
 The geometry of the word problem, Warwick Junior Geometry and Topology Seminar, 8 January 2018.
 Onerelator groups with no bad subgroups, Bielefeld Oberseminar Groups and Geometry, 3 January 2018.
 Onerelator groups with no bad subgroups, Technion Groups, Dynamics and Related Topics Seminar, 7 December 2017.
 Determining hyperbolic 3manifold groups by their finite quotients, Sydney Group Actions Seminar, 23 August 2017.
 Deficiencies of finite groups, yGAGTA, 28 June 2017. Talk notes.
 Hyperbolic groups and their subgroups, Glasgow Algebra Seminar, 15 February 2017
 The word problem and onerelator groups, Cambridge Geometric Group Theory Seminar, 10 August 2016
Other writing

My doctoral thesis Encoding and Detecting Properties in Finitely Presented Groups (available for posterity via permalink at the Oxford University Research Archive)

Expander Graphs and Kazhdan's Property (T), my 2012 Honours thesis at the University of Sydney

I made some posters
Teaching
University of Münster
 SS 2021: seminar on representation theory of finite groups for the dualsubject bachelor's programme, with Arthur Bartels
 WS 2020/21: Repetitorium, lecture course revising first year linear algebra and analysis in preparation for oral exams (Learnweb site)
 WS 2020/21: seminar on profinite groups, with Tim Clausen (Learnweb site)
 SS 2020: seminar on hyperbolic groups, with Arthur Bartels (Learnweb site)
 WS 2019/20: assistant for Topologie I (Learnweb site)
 WS 2019/20: Repetitorium (Learnweb site)
 SS 2019: assistant for Mathematik für Physiker II (Learnweb site)
 SS 2019: seminar on matrix groups for the dualsubject bachelor's programme, with Johannes Ebert (QISPOS entry)
University of Oxford
 Hilary 2017: Tutor for C3.2 Geometric Group Theory
TA for C3.9 Computational Algebraic Topology  Michaelmas 2016: Tutor for B3.5 Topology and Groups
 Hilary 2016: Tutor for C3.2 Geometric Group Theory
 Michaelmas 2015: Tutor for 2x B3.5 Topology and Groups
 Michaelmas 2014: Tutor for 2x B3.5 Topology and Groups
University of Sydney
 Semester 1 2013: Tutor for 4x MATH2069 / MATH2969 Discrete Mathematics and Graph Theory
Tutor for INFO1903 Informatics (Advanced)  Summer School Semester 2013: Lecturer and Tutor for MATH1004 Discrete Mathematics
 Semester 1 2012: Tutor for 2x MATH1901 Differential Calculus (Advanced)
Where you might meet me
 No upcoming travel plans
Where you might have met me
 Outer space in Bielefeld, Bielefeld, 2426 September 2019
 Ventotene International Workshops "Of coarse!", Ventotene, 914 September 2019
 Flags, Galleries and Reflection Groups, Sydney, 59 August 2019
 YGGT VIII, Bilbao, 30 June5 July 2019
 SPP Conference "Geometry at Infinity", Münster, 15 April 2019
 Geometric Group Theory in Bonn, MPIM Bonn, 31 January1 February 2019
 Equations in OneRelator Groups and Semigroups, ICMS Edinburgh, 37 September 2018
 Nonpositively Curved Groups on the Mediterranean, Nahsholim, 2329 May 2018
 THGT 2018, Bielefeld, 36 April 2018
 YGGT VII, Les Diablerets, 1216 March 2018
 Winter OneRelator Workshop, University of East Anglia, Norwich, 911 January 2018
 Ventotene International Workshops "Moduli Spaces", Ventotene, 1116 September 2017
 yGAGTA, Bilbao, 2630 June 2017
 YGGT VI, Oxford, 2024 March 2017
 Nonpositive curvature in action, INI Cambridge, 913 January 2017
 Introductory Workshop: Geometric Group Theory, MSRI Berkeley, 2226 August 2016
 Computation in geometric and combinatorial group theory, ICMS Edinburgh, 1115 July 2016
 Beyond Hyperbolicity, Cambridge, 2024 June 2016
 YGGT V, KIT Karlsruhe, 1519 February 2016
 GAGTA, CIRM Luminy, 1418 September 2015
 Conference on Topology and Geometry, HCM Bonn, 1721 August 2015
 Growth, Symbolic Dynamics and Combinatorics of Words in Groups, ENS Paris, 15 June 2015
 YGGT IV, Spa, January 2015
 Cube Complexes and Groups, SYM Copenhagen, 711 July 2014
 Asymptotic properties of groups, IHP Paris, 2428 March 2014
 Geometry and Groups after Thurston TCD Dublin, 2731 August 2013
 Surface subgroups and cube complexes, MSRI Berkeley, 1822 March 2013
 Groups and Geometry in the South East, South East England, various meetings 20132017