Stable equivalence of bridge positions of a handlebody-knot

We show that any two bridge positions of a handlebody-knot are stably equivalent.

Brauer indecomposability of Scott modules with semidihedral vertex

We present a sufficient condition for the $kG$ -Scott module with vertex $P$ to remain indecomposable under the Brauer construction for any subgroup $Q$ of $P$ as $k[Q\,C_G(Q)]$ -module, where $k$ is a field of characteristic $2$ , and $P$ is a semidihedral $2$ -subgroup of a finite group $G$ . This generalizes results for the cases where $P$ is abelian or dihedral. The Brauer indecomposability is defined by R. Kessar, N. Kunugi and N. Mitsuhashi. The motivation of this paper is the fact that the Brauer indecomposability of a $p$ -permutation bimodule (where $p$ is a prime) is one of the key steps in order to obtain a splendid stable equivalence of Morita type by making use of the gluing method due to Broué, Rickard, Linckelmann and Rouquier, that then can possibly be lifted to a splendid derived (splendid Morita) equivalence.

Validation of automatic passenger counting: introducing the t-test-induced equivalence test

Automatic passenger counting (APC) in public transport has been introduced in the 1970s and has been rapidly emerging in recent years. Still, real-world applications continue to face events that are difficult to classify. The induced imprecision needs to be handled as statistical noise and thus methods have been defined to ensure that measurement errors do not exceed certain bounds. Various recommendations for such an APC validation have been made to establish criteria that limit the bias and the variability of the measurement errors. In those works, the misinterpretation of non-significance in statistical hypothesis tests for the detection of differences (e.g. Student’s t-test) proves to be prevalent, although existing methods which were developed under the term equivalence testing in biostatistics (i.e. bioequivalence trials, Schuirmann in J Pharmacokinet Pharmacodyn 15(6):657–680, 1987) would be appropriate instead. This heavily affects the calibration and validation process of APC systems and has been the reason for unexpected results when the sample sizes were not suitably chosen: Large sample sizes were assumed to improve the assessment of systematic measurement errors of the devices from a user’s perspective as well as from a manufacturers perspective, but the regular t-test fails to achieve that. We introduce a variant of the t-test, the revised t-test, which addresses both type I and type II errors appropriately and allows a comprehensible transition from the long-established t-test in a widely used industrial recommendation. This test is appealing, but still it is susceptible to numerical instability. Finally, we analytically reformulate it as a numerically stable equivalence test, which is thus easier to use. Our results therefore allow to induce an equivalence test from a t-test and increase the comparability of both tests, especially for decision makers.

Special moves for open book decompositions of 3-manifolds

We provide a complete set of two moves that suffice to relate any two open book decompositions of a given 3-manifold. One of the moves is the usual plumbing with a positive or negative Hopf band, while the other one is a special local version of Harer’s twisting, which is presented in two different (but stably equivalent) forms. Our approach relies on 4-dimensional Lefschetz fibrations, and on 3-dimensional contact topology, via the Giroux-Goodman stable equivalence theorem for open book decompositions representing homologous contact structures.

Stability of products of equivalence relations

An ergodic probability measure preserving (p.m.p.) equivalence relation ${\mathcal{R}}$ is said to be stable if ${\mathcal{R}}\cong {\mathcal{R}}\times {\mathcal{R}}_{0}$ where ${\mathcal{R}}_{0}$ is the unique hyperfinite ergodic type $\text{II}_{1}$ equivalence relation. We prove that a direct product ${\mathcal{R}}\times {\mathcal{S}}$ of two ergodic p.m.p. equivalence relations is stable if and only if one of the two components ${\mathcal{R}}$ or ${\mathcal{S}}$ is stable. This result is deduced from a new local characterization of stable equivalence relations. The similar question on McDuff $\text{II}_{1}$ factors is also discussed and some partial results are given.

Külshammer ideals of algebras of quaternion type

For a symmetric algebra [Formula: see text] over a field [Formula: see text] of characteristic [Formula: see text] Külshammer constructed a descending sequence of ideals of the center of [Formula: see text]. If [Formula: see text] is perfect, this sequence was shown to be an invariant under derived equivalence and for algebraically closed [Formula: see text] the dimensions of their image in the stable center were shown to be invariant under stable equivalence of Morita type. Erdmann classified algebras of tame representation type which may be blocks of group algebras, and Holm classified Erdmann’s list up to derived equivalence. In both classifications, certain parameters occur in the classification, and it was unclear if different parameters lead to different algebras. Erdmann’s algebras fall into three classes, namely of dihedral, semidihedral and of quaternion type. In previous joint work with Holm, we used Külshammer ideals to distinguish classes with respect to these parameters in case of algebras of dihedral and semidihedral type. In the present paper, we determine the Külshammer ideals for algebras of quaternion type and distinguish again algebras with respect to certain parameters.

Inner amenability, property Gamma, McDuff factors and stable equivalence relations

We say that a countable group $G$ is McDuff if it admits a free ergodic probability measure preserving action such that the crossed product is a McDuff $\text{II}_{1}$ factor. Similarly, $G$ is said to be stable if it admits such an action with the orbit equivalence relation being stable. The McDuff property, stability, inner amenability and property Gamma are subtly related and several implications and non-implications were obtained in Effros [Property $\unicode[STIX]{x1D6E4}$ and inner amenability. Proc. Amer. Math. Soc.47 (1975), 483–486], Jones and Schmidt [Asymptotically invariant sequences and approximate finiteness. Amer. J. Math.109 (1987), 91–114], Vaes [An inner amenable group whose von Neumann algebra does not have property Gamma. Acta Math.208 (2012), 389–394], Kida [Inner amenable groups having no stable action. Geom. Dedicata173 (2014), 185–192] and Kida [Stability in orbit equivalence for Baumslag–Solitar groups and Vaes groups. Groups Geom. Dyn.9 (2015), 203–235]. We complete the picture with the remaining implications and counterexamples.