Simple groups separated by finiteness properties

We show that for every positive integer $n$ there exists a simple group that is of type $\mathrm{F}_{n-1}$ but not of type $\mathrm{F}_n$. For $n\ge 3$ these groups are the first known examples of this kind. They also provide infinitely many quasi-isometry classes of finitely presented simple groups. The only previously known infinite family of such classes, due to Caprace--R\'emy, consists of non-affine Kac--Moody groups over finite fields. Our examples arise from R\"over--Nekrashevych groups, and contain free abelian groups of infinite rank.Comment: 25 pages. v2: incorporated comments v3: final version, to appear, Invent. Mat

Concatenated dynamical decoupling in a solid-state spin bath

Concatenated dynamical decoupling (CDD) pulse sequences hold much promise as a strategy to mitigate decoherence in quantum information processing. It is important to investigate the actual performance of these dynamical decoupling strategies in real systems that are promising qubit candidates. In this Rapid Communication, we compute the echo decay of concatenations of the Hahn echo sequence for a solid-state electronic spin qubit in a nuclear spin bath using a cluster expansion technique. We find that each level of concatenation reverses the effect of successive levels of intrabath fluctuations. On the one hand, this advances CDD as a versatile and realistic decoupling strategy. On the other hand, this invalidates, as overly optimistic, results of the simple pair approximation used previously to study restoration, through CDD, of coherence lost to a mesoscopic spin bath