# Sam Hughes

### PhD Student, University of Southampton

Since October 2018 I have been a PhD student at the University of Southampton under the supervision of Professor Ian Leary. Before I became a student at Southampton, I completed an MMath degree at the University of South Wales. In October 2021 I will be joining Dr Dawid Kielak's team in the Mathematical Institute at the University of Oxford

My research is primarily in geometric group theory, often with a topological flavour. More specifically, I am interested in lattices in locally compact groups, cohomology of arithmetic groups, $$L^2$$-invariants, and $$K$$-theory in relation to the isomorphism conjectures.

My recent activities can be found here.

I was one of the organisers of the interdisciplinary seminar series Maths and Mingle. You can read more about Maths and Mingle in our article in the May 2020 edition of the LMS newsletter. I am also co-organising the conference Postgraduate Group Theory 2020 (PGTC 2020).

## Preprints and publications

Click titles for abstracts.
1. Graphs and complexes of lattices, in preparation.
Poster, slides.
In preparation.

2. (With Indira L. Chatterji and Peter H. Kropholler) Groups acting on trees and the first $$\ell^2$$-Betti number, submitted.
arXiv, pdf.
We generalise results of Thomas, Allcock, Thom-Petersen, and Kar-Niblo to the first $$\ell^2$$-Betti number of quotients of certain groups acting on trees by subgroups with free actions on the edge sets of the graphs.

3. On the equivariant $$K$$- and $$KO$$-homology of some special linear groups, to appear in Algebraic and Geometric Topology.
arXiv, pdf, slides.
We compute the equivariant $$KO$$-homology of the classifying space for proper actions of $$\textrm{SL}_3(\mathbb{Z})$$ and $$\textrm{GL}_3(\mathbb{Z})$$. We also compute the Bredon homology and equivariant $$K$$-homology of the classifying spaces for proper actions of $$\textrm{PSL}_2(\mathbb{Z}[\frac{1}{p}])$$ and $$\textrm{SL}_2(\mathbb{Z}[\frac{1}{p}])$$ for each prime $$p$$. Finally, we prove the Unstable Gromov-Lawson-Rosenberg Conjecture for a large class of groups whose maximal finite subgroups are odd order and have periodic cohomology.

4. Cohomology of Fuchsian groups and non-Euclidean crystallographic groups, submitted.
arXiv, pdf.
For each geometrically finite non-Euclidean crystallographic group (NEC group), we compute the cohomology groups. In the case where the group is a Fuchsian group, we also determine the ring structure of the cohomology. Finally, we compute the $$L^2$$-Betti numbers of the NEC groups.

5. (With Nick Gill) The character table of a sharply $$5$$-transitive subgroup of the alternating group of degree $$12$$, to appear in International Journal of Group Theory.
Journal, arXiv, pdf.
In this paper we calculate the character table of a sharply $$5$$-transitive subgroup of $${\rm Alt}(12)$$, and of a sharply $$4$$-transitive subgroup of $$\rm{Alt}(11$$). Our presentation of these calculations is new because we make no reference to the sporadic simple Mathieu groups, and instead deduce the desired character tables using only the existence of the stated multiply transitive permutation representations.

## Contact

Email: sam.hughes@soton.ac.uk
Office: Room 9007, Building 54, Highfield Campus

My institutional page is here.
ResearchGate. Scholar.