36 citations · 50 across the 10 of their papers we have counts for
12 papers
Non-trivial units of complex group rings
Giles Gardam
The Kaplansky unit conjecture for group rings is false in characteristic zero.
Graphs and groups with unique geodesics
Murray Elder, Giles Gardam, Adam Piggott +2
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 g…
The Surface Group Conjectures for groups with two generators
Giles Gardam, Dawid Kielak, Alan D. Logan
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…
Algebraically hyperbolic groups
Giles Gardam, Dawid Kielak, Alan D. Logan
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…
A counterexample to the unit conjecture for group rings
Giles Gardam
The unit conjecture, commonly attributed to Kaplansky, predicts that if is a field and is a torsion-free group then the only units of the group ring are the trivial…
JSJ decompositions and polytopes for two-generator one-relator groups
Giles Gardam, Dawid Kielak, Alan D. Logan
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 abelianis…
Two generalisations of Leighton's Theorem
Sam Shepherd, Giles Gardam, Daniel J. Woodhouse
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 mode…
Cannon--Thurston maps for CAT(0) groups with isolated flats
Benjamin Beeker, Matthew Cordes, Giles Gardam +2
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 hype…
Profinite rigidity in the SnapPea census
Giles Gardam
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.…
The geometry of one-relator groups satisfying a polynomial isoperimetric inequality
Giles Gardam, Daniel J. Woodhouse
For every pair of positive integers we construct a one-relator group whose Dehn function is where . The group has no…
Detecting laws in power subgroups
Giles Gardam
A group law is said to be detectable in power subgroups if, for all coprime and , a group satisfies the law if and only if the power subgroups and both satis…
Finite groups of arbitrary deficiency
Giles Gardam
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 deficie…