Office 1.23,

Fry Building,

Woodland Road,

Bristol,

BS8 1TH

- (With Thomas Haettel and Nima Hoda)
*The coarse Helly property, hierarchical hyperbolicity, and semihyperbolicity*. 33 pages.`arxiv:2009.14053`.Abstract .

We relate three classes of nonpositively curved metric spaces: hierarchically hyperbolic spaces, coarsely Helly spaces, and strongly shortcut spaces. We show that any hierarchically hyperbolic space admits a new metric that is coarsely Helly. The new metric is quasi-isometric to the original metric and is preserved under automorphisms of the hierarchically hyperbolic space. We show that any coarsely Helly metric space of uniformly bounded geometry is strongly shortcut. Consequently, hierarchically hyperbolic groups---including mapping class groups of surfaces---are coarsely Helly and coarsely Helly groups are strongly shortcut.

Using these results we deduce several important properties of hierarchically hyperbolic groups, including that they are semihyperbolic, have solvable conjugacy problem, are of type $FP_{\infty}$, have finitely many conjugacy classes of finite subgroups, and their finitely generated abelian subgroups are undistorted. Along the way we show that hierarchically quasiconvex subgroups of hierarchically hyperbolic groups have bounded packing. - (With Davide Spriano)
*Unbounded domains in hierarchically hyperbolic groups*. Submitted, 18 pages.`arxiv:2007.12535`.Abstract .

We investigate unbounded domains in hierarchically hyperbolic groups and obtain constraints on the possible hierarchical structures. Using these insights, we characterise the structures of virtually abelian HHGs and show that the class of HHGs is not closed under finite extensions. This provides a strong answer to the question of whether being an HHG is invariant under quasiisometries. Along the way, we show that infinite torsion groups are not HHGs.

By ruling out pathological behaviours, we are able to give simpler, direct proofs of the rank-rigidity and omnibus subgroup theorems for HHGs. This involves extending our techniques so that they apply to all subgroups of HHGs. - (With Mark Hagen)
*Projection complexes and quasimedian maps*. Submitted, 19 pages. Preprint.Abstract .

We use the projection complex machinery of Bestvina–-Bromberg-–Fujiwara to study hierarchically hyperbolic groups. In particular, we show that if the associated hyperbolic spaces are quasiisometric to trees then the group is quasiisometric to a finite dimensional CAT(0) cube complex. We use this to deduce a number of properties, including the Helly property for hierarchically quasiconvex subsets, and the fact that such a group has finitely many conjugacy classes of finite subgroups. Future work will examine the extent to which the quasitree hypothesis can be replaced by more straightforward algebraic assumptions, in the presence of a BBF colouring.

*The special linear group for nonassociative rings*. J. Group Theory 23 (2020), no. 2, 327-335.`arxiv:1807:05227`; journal link.Abstract .

We extend to arbitrary rings a definition of the octonion special linear group due to Baez. At the infinitesimal level we get a Lie ring, which we describe over some large classes of rings, including all associative rings and all algebras over a field. As a corollary we compute all the groups Baez defined.*Derivations of octonion matrix algebras*. Comm. Algebra 47 (2019), no. 10, 4216-4223.`arxiv:1809.02222`; journal link.Abstract .

It is well-known that the exceptional Lie algebras $\mathfrak{f}_4$ and $\mathfrak{g}_2$ arise from the octonions as the derivation algebras of the $3\times3$ hermitian and $1\times1$ antihermitian matrices, respectively. Inspired by this, we compute the derivation algebras of the spaces of hermitian and antihermitian matrices over an octonion algebra in all dimensions.

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