arXiv subjects

A. Khukhro

Publications and source records attributed to A. Khukhro.

5 recordsLinked to original sources

arXiv2020

A characterisation of virtually free groups via minor exclusion

We give a new characterisation of virtually free groups using graph minors. Namely, we prove that a finitely generated, infinite group is virtually free if and only if for any finite generating set, the corresponding Cayley graph is minor excluded. This answers a question of Ostrovskii and Rosenthal. The proof relies on showing that a finitely generated group that is minor excluded with respect to every finite generating set is accessible, using a graph-theoretic characterisation of accessibility due to Thomassen and Woess.

math.GR↗

arXiv2013

Embeddable box spaces of free groups

We generalize the construction of Arzhantseva, Guentner and Spakula of a box space of the free group which admits a coarse embedding into Hilbert space. We show that for a finitely generated free group, the box space corresponding to the derived $m$-series (for any integer $m\geq 2$) coarsely embeds into Hilbert space. This gives new examples of metric spaces with bounded geometry which coarsely embed into Hilbert space but do not have Yu's property A.

math.GR↗

arXiv2011

Box spaces, group extensions and coarse embeddings into Hilbert space

We investigate how coarse embeddability of box spaces into Hilbert space behaves under group extensions. In particular, we prove a result which implies that a semidirect product of a finitely generated free group by a finitely generated residually finite amenable group has a box space which coarsely embeds into Hilbert space. This provides a new class of examples of metric spaces with bounded geometry which coarsely embed into Hilbert space but do not have property A, generalising the example of Arzhantseva, Guentner and Spakula.

math.GR↗

arXiv2011

Expanders and Property A

We give a cohomological characterisation of expander graphs, and use it to give a direct proof that expander graphs do not have Yu's property A.

math.GT↗

arXiv2011

Uniform Embeddability into Hilbert Space

The open question of what prevents a metric space with bounded geometry from being uniformly embeddable in Hilbert space is answered here for box spaces of residually finite groups. We prove that a box space does not contain a uniformly embedded expander sequence if and only if it uniformly embeds in Hilbert space. In particular, this gives a sufficient condition for a residually finite group to have the Haagerup property. The main result holds in the more general setting of a disjoint union of Cayley graphs of finite groups with bounded degree.

math.GR↗