# 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>@uni-muenster.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 non-positive curvature (for instance, in what settings are subgroups like Baumslag-Solitar groups the only obstruction?), group rings, one-relator 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 talk

The unit conjecture, commonly attributed to Kaplansky, predicts that if $$K$$ is a field and $$G$$ is a torsion-free group then the only units of the group ring $$K[G]$$ are the trivial units, that is, the non-zero 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 two-generator one-relator 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 two-generator one-relator group with abelianisation of rank $$2$$. We deduce various structural and algorithmic properties, like the existence of a quadratic-time 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), 1100–1104

The 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 non-positive deficiency. We show that every non-positive 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 one-relator groups satisfying a polynomial isoperimetric inequality with Daniel Woodhouse, arXiv:1711.08755 journal version
Proc. Amer. Math. Soc. 147 (2019), 125–129

For every pair of positive integers $$p > q$$ we construct a one-relator 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 Baumslag-Solitar 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 long-standing question on the automaticity of one-relator 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 well-known question asks whether any two non-isometric finite volume hyperbolic 3-manifolds 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 once-punctured 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 3-manifolds have the same finite quotients.

• Minimal sizeable graphs, arXiv:1808.09505 appendix to the paper Hyperbolic Groups with Finitely Presented Subgroups not of Type F3 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 torsion-free normal hyperbolic subgroups of a non-hyperbolic CAT(0) group with isolated flats with respect to the visual boundaries. We also show Cannon–Thurston maps do not exist for infinite torsion-free infinite-index normal CAT(0) subgroups with isolated flats in a non-hyperbolic CAT(0) group with isolated flats. We determine the isomorphism types of the normal subgroups in the above two settings.

• Alternative proof of ball-restricted version, arXiv:1908.00830 appendix to the paper Two generalisations of Leighton's Theorem of S. Shepherd, joint with Daniel Woodhouse (submitted)

• ...

## Teaching

### University of Münster

• SS 2021: seminar on representation theory of finite groups for the dual-subject 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 dual-subject bachelor's programme, with Johannes Ebert (QISPOS entry)
I had no teaching duties in WS 2018/19 in Münster or at the Technion.

## Where you might meet me

• No upcoming travel plans