The Surface Group Conjectures for groups with two generators

Funding: This work has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant agreement No. 850930), and from the Engineering and Physical Sciences Research Council (EPSRC), grants EP/R035814/1 and EP/S010963/1, and was supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – Project ID 427320536 – SFB 1442, as well as under Germany’s Excellence Strategy EXC 2044–390685587, Mathematics Münster: Dynamics–Geometry–Structure.The Surface Group Conjectures are statements about recognising surface groups among one-relator groups, using either the structure of their finite-index subgroups, or all subgroups. We resolve these conjectures in the two generator case. More generally, we prove that every two-generator one-relator group with every infinite-index subgroup free is itself either free or a surface group.PostprintPeer reviewe

Accompanying code to "Non-trivial units of complex group rings"

<p>This repository is for code to accompany the paper "<a href="https://arxiv.org/abs/2312.05240">Non-trivial units of complex group rings</a>". At the moment it is just 3 scripts, verifying the 2 theorems and a group presentation computation mentioned in the paper.</p&gt

Encoding and detecting properties in finitely presented groups

In this thesis we study several properties of finitely presented groups, through the unifying paradigm of encoding sought-after group properties into presentations and detecting group properties from presentations, in the context of Geometric Group Theory. 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. We construct a census of two-generator one-relator groups of relator length at most 9, with complete determination of isomorphism type, and verify a conjecture regarding conditions under which such groups are automatic. Furthermore, we introduce a family of one-relator groups and classify which of them act properly cocompactly on complete CAT(0) spaces; the non-CAT(0) examples are counterexamples to a variation on the aforementioned conjecture. For a subclass, we establish automaticity, which is needed for the census. 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. For every prime p we construct finite p-groups of arbitrary negative deficiency, and thereby complete Kotschick’s proposed classification of the integers which are deficiencies of Kähler groups. We explore variations and embellishments of our basic construction, which require subtle Schur multiplier computations, and we investigate the conditions on inputs to the construction that are necessary for success. 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. We determine examples of sizeable graphs, as required to construct finitely presented non-hyperbolic subgroups of hyperbolic groups, which have the fewest vertices possible modulo mild topological assumptions.</p

Media Representations of Policies Concerning Education Access and their Roles in Seven First Nations Students’ Deaths in Northern Ontario

We employed postcolonial theory, a case study methodology, and critical discourse analysis to investigate the ways in which non-First Nations and First Nations news sources produced understandings of the role(s) that education policies may have played in the deaths of seven First Nations students in Thunder Bay, Ontario, Canada. We found that national non-First Nations media sources produced the discourse that First Nations peoples require federal government policy as a form of intervention in their lives. Further, we found that though these media sources focused on criticizing the present state of First Nations education, they ignored the colonial processes and policies that contributed to a situation that resulted in the students attending high school in Thunder Bay, rather than their home communities. First Nations and local (Thunder Bay) non-First Nations media sources, however, emphasized the need for greater cooperation between the Canadian government and First Nations peoples to resolve the long-standing policy issues that continue to affect First Nations youth and their education in northern Ontario. These findings point to important differences in the ways in which various forms of media cover First Nations policy issues