Giles Gardam
I am a postdoc working in geometric group theory at the Technion. I will participate in the Hausdorff Trimester Program Logic and Algorithms in Group Theory from September to December 2018, and from January 2019 I will be working in the Topology Group at the WWU Münster as a Research Associate (Akademischer Rat auf Zeit). My graduate studies were supervised by Martin Bridson in Oxford. I did my undergraduate studies at the University of Sydney and wrote my honours thesis under the supervision of Anne Thomas.
I successfully defended my DPhil thesis on 1 September 2017.
You can contact me at
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?), onerelator groups, 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
I have an arXiv page.

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.

Detecting laws in power subgroups, arXiv:1705.09348
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.

The geometry of onerelator groups satisfying a polynomial isoperimetric inequality with Daniel Woodhouse, arXiv:1711.08755
For 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.

Profinite rigidity in the SnapPea census, arXiv:1805.02697
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, appendix to the paper Hyperbolic Groups with Finitely Presented Subgroups not of Type F_{3} of R. Kropholler, in preparation (available on request)

Onerelator groups are not virtually residually solvable with Dawid Kielak, in preparation
In this note we show that finitely generated groups with all finite quotients abelian, such as Baumslag's onerelator group, are not virtually residually solvable. This is motivated by the question of whether all onerelator groups have the Haagerup property.
Upcoming talks
 Neuchâtel, 15 May 2018
 Lausanne, 17 May 2018
Recent talks
 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
Teaching
I have no teaching duties at the Technion.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)
Posters
I made a poster titled The word problem and onerelator groups (NB: 2.5 MB TIFF file, or click thumbnail below for 740 KB JPEG). It outlines a project in which I verified that a problem suggested by Myasnikov, Ushakov and Won has a positive answer for many small cases. My paper with Daniel Woodhouse shows that the answer is false in general.
There's also a poster corresponding to an old version of the paper Finite groups of arbitrary deficiency (a 3.0 MB PDF file, or 700 KB JPEG linked from thumbnail). The new version avoids the spectral sequence computation.
Where you might meet me
 Nonpositively Curved Groups on the Mediterranean, Nahsholim, 2329 May 2018
Where you might have met me
 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