Weak equivalence and non-classifiability of measure preserving actions

Abért and Weiss have shown that the Bernoulli shift s_Γ of a countably infinite group Γ is weakly contained in any free measure preserving action ɑ of Γ. Proving a conjecture of Ioana, we establish a strong version of this result by showing that s_Γ×ɑ is weakly equivalent to ɑ. Using random Bernoulli shifts introduced by Abért, Glasner, and Virag, we generalize this to non-free actions, replacing s_Γ with a random Bernoulli shift associated to an invariant random subgroup, and replacing the product action with a relatively independent joining. The result for free actions is used along with the theory of Borel reducibility and Hjorth’s theory of turbulence to show that, on the weak equivalence class of a free measure preserving action, the equivalence relations of isomorphism, weak isomorphism, and unitary equivalence are not classifiable by countable structures. This in particular shows that there are no free weakly rigid actions, that is, actions whose weak equivalence class and isomorphism class coincide, answering negatively a question of Abért and Elek. We also answer a question of Kechris regarding two ergodic theoretic properties of residually finite groups. A countably infinite residually finite group Γ is said to have property EMD∗ if the action p_Γ of Γ on its profinite completion weakly contains all ergodic measure preserving actions of Γ, and Γ is said to have property MD if ι×p_Γ weakly contains all measure preserving actions of Γ, where ι denotes the identity action on a standard non-atomic probability space. Kechris has shown that EMD∗ implies MD and asked if the two properties are actually equivalent. We provide a positive answer to this question by studying the relationship between convexity and weak containment in the space of measure preserving actions

Ultraproducts of measure preserving actions and graph combinatorics

Ultraproducts of measure preserving actions of countable groups are used to study the graph combinatorics associated with such actions, including chromatic, independence and matching numbers. Applications are also given to the theory of random colorings of Cayley graphs and sofic actions and equivalence relations

Blood-based epigenome-wide analyses of cognitive abilities

BACKGROUND: Blood-based markers of cognitive functioning might provide an accessible way to track neurodegeneration years prior to clinical manifestation of cognitive impairment and dementia. RESULTS: Using blood-based epigenome-wide analyses of general cognitive function, we show that individual differences in DNA methylation (DNAm) explain 35.0% of the variance in general cognitive function (g). A DNAm predictor explains ~4% of the variance, independently of a polygenic score, in two external cohorts. It also associates with circulating levels of neurology- and inflammation-related proteins, global brain imaging metrics, and regional cortical volumes. CONCLUSIONS: As sample sizes increase, the ability to assess cognitive function from DNAm data may be informative in settings where cognitive testing is unreliable or unavailable. SUPPLEMENTARY INFORMATION: The online version contains supplementary material available at 10.1186/s13059-021-02596-5

Within-sibship genome-wide association analyses decrease bias in estimates of direct genetic effects

Estimates from genome-wide association studies (GWAS) of unrelated individuals capture effects of inherited variation (direct effects), demography (population stratification, assortative mating) and relatives (indirect genetic effects). Family-based GWAS designs can control for demographic and indirect genetic effects, but large-scale family datasets have been lacking. We combined data from 178,086 siblings from 19 cohorts to generate population (between-family) and within-sibship (within-family) GWAS estimates for 25 phenotypes. Within-sibship GWAS estimates were smaller than population estimates for height, educational attainment, age at first birth, number of children, cognitive ability, depressive symptoms and smoking. Some differences were observed in downstream SNP heritability, genetic correlations and Mendelian randomization analyses. For example, the within-sibship genetic correlation between educational attainment and body mass index attenuated towards zero. In contrast, analyses of most molecular phenotypes (for example, low-density lipoprotein-cholesterol) were generally consistent. We also found within-sibship evidence of polygenic adaptation on taller height. Here, we illustrate the importance of family-based GWAS data for phenotypes influenced by demographic and indirect genetic effects

Genomic Relationships, Novel Loci, and Pleiotropic Mechanisms across Eight Psychiatric Disorders

Genetic influences on psychiatric disorders transcend diagnostic boundaries, suggesting substantial pleiotropy of contributing loci. However, the nature and mechanisms of these pleiotropic effects remain unclear. We performed analyses of 232,964 cases and 494,162 controls from genome-wide studies of anorexia nervosa, attention-deficit/hyper-activity disorder, autism spectrum disorder, bipolar disorder, major depression, obsessive-compulsive disorder, schizophrenia, and Tourette syndrome. Genetic correlation analyses revealed a meaningful structure within the eight disorders, identifying three groups of inter-related disorders. Meta-analysis across these eight disorders detected 109 loci associated with at least two psychiatric disorders, including 23 loci with pleiotropic effects on four or more disorders and 11 loci with antagonistic effects on multiple disorders. The pleiotropic loci are located within genes that show heightened expression in the brain throughout the lifespan, beginning prenatally in the second trimester, and play prominent roles in neurodevelopmental processes. These findings have important implications for psychiatric nosology, drug development, and risk prediction.Peer reviewe

Mixing actions of countable groups are almost free

Abstract. A measure preserving action of a countably infinite group Γ is called totally ergodic if every infinite subgroup of Γ acts ergodically. For example, all mixing and mildly mixing actions are totally ergodic. This note shows that if an action of Γ is totally ergodic then there exists a finite normal subgroup N of Γ such that the stabilizer of almost every point is equal to N. Surprisingly the proof relies on the group theoretic fact (proved by Hall and Kulatilaka as well as by Kargapolov) that every infinite locally finite group contains an infinite abelian subgroup, of which all known proofs rely on the Feit-Thompson theorem. As a consequence we deduce a group theoretic characterization of countable groups whose non-trivial Bernoulli factors are all free: these are precisely the groups that posses no finite normal subgroup other than the trivial subgroup. 1

On a co-induction question of Kechris

This note answers a question of Kechris: if H < G is a normal subgroup of a countable group G, H has property MD and G/H is amenable and residually finite, then G also has property MD. Under the same hypothesis we prove that for any action a of G, if b is a free action of G/H, and b_G is the induced action of G, then CInd^G_H (ɑ|H) × b_G weakly contains ɑ. Moreover, if H < G is any subgroup of a countable group G, and the action of G on G/H is amenable, then CInd^G_H (ɑ|H) weakly contains ɑ whenever ɑ is a Gaussian action