New uniform diameter bounds in pro-pp groups

We give new upper bounds for the diameters of finite groups which do not depend on a choice of generating set. Our method exploits the commutator structure of certain profinite groups, in a fashion analogous to the Solovay–Kitaev procedure from quantum computation. We obtain polylogarithmic upper bounds for the diameters of finite quotients of groups with an analytic structure over a pro-p domain (with exponent depending on the dimension); Chevalley groups over a pro-p domain (with exponent independent of the dimension) and the Nottingham group of a finite field. We also discuss some consequences of our results for random walks on groups

Local permutation stability

We introduce a notion of "local stability in permutations" for finitely generated groups. If a group is sofic and locally stable in our sense, then it is also locally embeddable into finite groups (LEF). Our notion is weaker than the "permutation stability" introduced by Glebsky-Rivera and Arzhantseva-Paunescu, which allows one to upgrade soficity to residual finiteness. We prove a necessary and sufficient condition for an amenable group to be locally permutation stable, in terms of invariant random subgroups (IRSs), inspired by a similar criterion for permutation stability due to Becker, Lubotzky and Thom. We apply our criterion to prove that derived subgroups of topological full groups of Cantor minimal subshifts are locally stable, using Zheng's classification of IRSs for these groups. This last result provides continuum-many groups which are locally stable, but not stable.Comment: 33 pages, comments welcome

Cool Core Bias in Sunyaev-Zel'dovich Galaxy Cluster Surveys

Sunyaev-Zeldovich (SZ) surveys find massive clusters of galaxies by measuring the inverse Compton scattering of cosmic microwave background off of intra-cluster gas. The cluster selection function from such surveys is expected to be nearly independent of redshift and cluster astrophysics. In this work, we estimate the effect on the observed SZ signal of centrally-peaked gas density profiles (cool cores) and radio emission from the brightest cluster galaxy (BCG) by creating mock observations of a sample of clusters that span the observed range of classical cooling rates and radio luminosities. For each cluster, we make simulated SZ observations by the South Pole Telescope and characterize the cluster selection function, but note that our results are broadly applicable to other SZ surveys. We find that the inclusion of a cool core can cause a change in the measured SPT significance of a cluster between 0.01% - 10% at z > 0.3, increasing with cuspiness of the cool core and angular size on the sky of the cluster (i.e., decreasing redshift, increasing mass). We provide quantitative estimates of the bias in the SZ signal as a function of a gas density cuspiness parameter, redshift, mass, and the 1.4 GHz radio luminosity of the central AGN. Based on this work, we estimate that, for the Phoenix cluster (one of the strongest cool cores known), the presence of a cool core is biasing the SZ significance high by ~ 6%. The ubiquity of radio galaxies at the centers of cool core clusters will offset the cool core bias to varying degrees.Comment: 8 pages, 4 figures, accepted to Ap

Short Laws for Finite Groups of Lie Type

We produce new short laws in two variables valid in finite groups of Lie type. Our result improves upon results of Kozma and the second named author, and is sharp up to logarithmic factors, for all families except possibly the Suzuki groups. We also produce short laws valid for generating pairs and random pairs in finite groups of Lie type, and, conditional on Babai's diameter conjecture, make effective the dependence of our bounds on the rank. Our proof uses, among other tools, the Classification of Finite Simple Groups, Aschbacher's structure theorem for maximal subgroups for classical groups, and upper bounds on the diameters of finite simple groups due to Breuillard, Green, Guralnick, Pyber, Szabo and Tao.Comment: 38 pages, comments welcom

Short laws for finite groups and residual finiteness growth

We prove that for every n ∈ N n \in \mathbb {N} and δ &gt; 0 \delta &gt;0 there exists a word w n ∈ F 2 w_n \in F_2 of length O ( n 2 / 3 log ⁡ ( n ) 3 + δ ) O(n^{2/3} \log (n)^{3+\delta }) which is a law for every finite group of order at most n n . This improves upon the main result of Andreas Thom [Israel J. Math. 219 (2017), pp. 469–478] by the second named author. As an application we prove a new lower bound on the residual finiteness growth of non-abelian free groups.</p