arXiv subjects
Publications and source records attributed to Giles Gardam.
We provide a direct connection between the 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 Z_{max}-JSJ decomposition of such groups.
We initiate the study of torsion-free algebraically hyperbolic groups; these groups generalise torsion-free hyperbolic groups and are intricately related to groups with no Baumslag--Solitar subgroups. Indeed, for groups of cohomological dimension $2$ we prove that algebraic hyperbolicity is equivalent to containing no Baumslag--Solitar subgroups. This links algebraically hyperbolic groups to two famous questions of Gromov; recent work has shown these questions to have negative answers in general, but they remain open for groups of cohomological dimension $2$. We also prove that algebraically hyperbolic groups are CSA, and so have canonical abelian JSJ-decompositions. In the two-generated case we give a precise description of the form of these decompositions.
A connected graph is called \emph{geodetic} if there is a unique geodesic between each pair of vertices. In this paper we prove that if a finitely generated group admits a Cayley graph which is geodetic, then the group must be virtually free. Before now, it was open whether finitely generated and geodetic implied hyperbolic. In fact we prove something more general: if a quasi-transitive locally finite connected undirected graph is geodetic then it is quasi-isometric to a tree. Our main tool is to define a \emph{boundary} of a graph and understand how the local behaviour influences it when the graph is geodetic. Our results unify, and represent significant progress on, research initiated by Ore, Shapiro, and Madlener and Otto.
The Kaplansky unit conjecture for group rings is false in characteristic zero.
Leighton's graph covering theorem says that two finite graphs with a common cover have a common finite cover. We present a new proof of this using groupoids, and use this as a model to prove two generalisations of the theorem. The first generalisation, which we refer to as the symmetry-restricted version, restricts how balls of a given size in the universal cover can map down to the two finite graphs when factoring through the common finite cover - this answers a question of Neumann. Secondly, we consider covers of graphs of spaces (or of more general objects), which leads to an even more general version of Leighton's Theorem. We also compute upper bounds for the sizes of the finite covers obtained in Leighton's Theorem and its generalisations. An appendix by Gardam and Woodhouse provides an alternative proof of the symmetry-restricted version, that uses Haar measure instead of groupoids.
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.
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.
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 normal hyperbolic subgroups of non-hyperbolic CAT(0) groups with isolated flats with respect to the visual boundaries. We also show Cannon--Thurston maps do not exist for infinite infinite-index normal CAT(0) subgroups with isolated flats in non-hyperbolic CAT(0) groups with isolated flats. We obtain a structure theorem for the normal subgroups in these settings and show that outer automorphism groups of hyperbolic groups have no purely atoroidal $\mathbb{Z}^2$ subgroups.
We generalise the constructions of Brady and Lodha to give infinite families of hyperbolic groups, each having a finitely presented subgroup that is not of type $F_3$. By calculating the Euler characteristic of the hyperbolic groups constructed, we prove that infinitely many of them are pairwise non isomorphic. We further show that the first of these constructions cannot be generalised to dimensions higher than $3$.
For every pair of positive integers $p > q$ we construct a one-relator group $R_{p,q}$ whose Dehn function is $\simeq n^{2 α}$ where $α= \log_2(2p / q)$. 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.
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.
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 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 Kaehler manifolds.