Journal of Topology and Analysis

In this paper, we first develop a general theory of Johnson filtrations and Johnson homomorphisms for a group $G$ acting on another group $K$ equipped with a filtration indexed by a “good” ordered commutative monoid. Then, specializing it to the case where the monoid is the additive monoid ${\mathbb{N}}^2$ of pairs on non-negative integers, we obtain a theory of double Johnson filtrations and homomorphisms. We apply this theory to the mapping class group $\mathcal{ℳ}$ of a surface ${\mathrm{\Sigma }}_{g,1}$ with one boundary component, equipped with the normal subgroups $\bar{X}$, $Ȳ$ of ${\pi }_1({\mathrm{\Sigma }}_{g,1})$ associated to a standard Heegaard splitting of the $3$-sphere. We also consider the case where the group $G$ is the automorphism group of a free group.

There are a number of known constructions of quasimorphisms on Hamiltonian groups. We show that on surfaces many of these quasimorphisms are not compatible with the Hofer norm in a sense they are not continuous and not Lipschitz. The only exception known to the author is the Calabi quasimorphism on a sphere [M. Entov and L. Polterovich, Calabi quasimorphism and quantum homology, Int. Math. Res. Not. 2003 (2003) 1635–1676] and induced quasimorphisms on genus-zero surfaces (e.g. [P. Biran, M. Entov and L. Polterovich, Calabi quasimorphisms for the symplectic ball, Commun. Contemp. Math. 6 (2004) 793–802]).

In this paper, we settle three basic questions concerning the Gutt–Hutchings capacities from [11] which are conjecturally equal to the Ekeland–Hofer capacities from [7, 8]. Our primary result settles a version of the recognition question from [4], in the negative. We prove that the Gutt–Hutchings capacities together with the volume do not constitute a complete set of symplectic invariants for star-shaped domains with smooth boundary. In particular, we construct a smooth family of such domains, all of which have the same Gutt–Hutchings capacities and volume, but no two of which are symplectomorphic. We also establish two independence properties of the Gutt–Hutchings capacities. We prove that, even for star-shaped domains with smooth boundaries, these capacities are independent from the volume by constructing a family of star-shaped domains with smooth boundaries, whose Gutt–Hutchings capacities all agree but whose volumes all differ. We also prove that the capacities are mutually independent by constructing, for any $j\in \mathbb{N}$, a family of star-shaped domains, with smooth boundary and the same volume, whose capacities are all equal but the $j$th. The constructions underlying these results are not exotic. They are convex and concave toric domains as defined in [11], where the authors also establish beautiful formulae for their capacities. A key to the progress made here is a significant simplification of these formulae under an additional symmetry assumption. This simplification allows us to identify new blind spots of the Gutt–Hutchings capacities which are used to construct the desired examples. This simplification also yields explicit computations of the Gutt–Hutchings capacities in many new examples.

Floer theory for Lagrangian cobordisms was developed by Biran and Cornea in a series of papers [Lagrangian cobordism. I, J. Amer. Math. Soc. 26 (2013) 295–340; Lagrangian cobordism and Fukaya categories, Geom. Funct. Anal. 24 (2014) 1731–1830; Cone-decompositions of Lagrangian cobordisms in Lefschetz fibrations, Selecta Math. 23 (2017) 2635–2704] to study the triangulated structure of the derived Fukaya category of monotone symplectic manifolds. This paper explains how to use the language of stops to study Lagrangian cobordisms in Liouville manifolds and the associated exact triangles in the derived wrapped Fukaya category. Furthermore, we compute the cobordism groups of non-compact Riemann surfaces of finite type.

We introduce a condition on the monodromy of a free-by-cyclic group, $G_{\varphi }$, called the chain flare condition, that implies that the $L^2$–torsion, ${\rho }^{(2)}(G_{\varphi })$, is nonzero. We conjecture that this condition holds whenever the monodromy is exponentially growing.

In this paper, we show that a sufficient and necessary condition for that the Hoeffding concentration inequality of Banach space-valued ($B$-valued, for simplicity) random variables holds is that the Banach space in question admits the Rademacher type $p$ for some $1\le p\le 2$; which is equivalent to that the Bernstein concentration inequality holds for such $B$-valued random variables. This is done by applying a refined McDiarmid inequality (stated and proved in this paper) via the entropy method. This result is also applied to the density property of induced subgraphs of random Cayley graphs generated by a finite abelian group.