The cheap embedding principle: Dynamical upper bounds for homology growth
We provide upper bounds for logarithmic torsion homology growth and Betti number growth of groups, phrased in the language of measured group theory.
math.AT↗
arXiv subjects
Clara Loeh
Publications and source records attributed to Clara Loeh.
Combination of open covers with $\pi_1$-constraints
Let~$G$ be a group and let~$\mathcal{F}$ be a family of subgroups of~$G$. The generalised Lusternik--Schnirelmann category~$\operatorname{cat}_\mathcal{F}(G)$ is the minimal cardinality of covers of~$BG$ by open subsets with fundamental group in~$\mathcal{F}$. We prove a combination theorem for~$\operatorname{cat}_\mathcal{F}(G)$ in terms of the stabilisers of contractible $G$-CW-complexes. As applications for the amenable category, we obtain vanishing results for the simplicial volume of gluings of manifolds (along not necessarily amenable boundaries) and of cyclic branched coverings. Moreover, we deduce an upper bound for Farber's topological complexity, generalising an estimate for amalgamated products of Dranishnikov--Sadykov.
math.AT↗
The algebraic cheap rebuilding property
We present an axiomatic approach to combination theorems for various homological properties of groups and, more generally, of chain complexes. Examples of such properties include algebraic finiteness properties, $\ell^2$-invisibility, $\ell^2$-acyclicity, lower bounds for Novikov--Shubin invariants, and vanishing of homology growth. As a key example, we introduce an algebraic version of Ab\'ert--Bergeron--Fr\k{a}czyk--Gaboriau's cheap rebuilding property that implies vanishing of torsion homology growth and fits into our axiomatic framework for combination theorems. In particular, we obtain that certain graphs of groups with amenable vertex groups and elementary amenable edge groups have vanishing torsion homology growth.
math.GR↗
Universal finite functorial semi-norms
Functorial semi-norms on singular homology measure the "size" of homology classes. A geometrically meaningful example is the $\ell^1$-semi-norm. However, the $\ell^1$-semi-norm is not universal in the sense that it does not vanish on as few classes as possible. We show that universal finite functorial semi-norms do exist on singular homology on the category of topological spaces that are homotopy equivalent to finite CW-complexes. Our arguments also apply to more general settings of functorial semi-norms.
math.CT↗
A quadratic lower bound for the number of minimal geodesics
A minimal geodesic on a Riemannian manifold is a geodesic defined on $\mathbb{R}$ that lifts to a globally distance minimizing curve on the universal covering. Bangert proved that there is a lower bound for the number of geometrically distinct minimal geodesics of closed Riemannian manifolds that is linear in the first Betti number, using the stable norm unit ball on the first homology. We refine this method to obtain a quadratic lower bound.
math.DG↗
Integral foliated simplicial volume and ergodic decomposition
We establish an integration formula for integral foliated simplicial volume along ergodic decompositions. This is analogous to the ergodic decomposition formula for the cost of groups.
math.GT↗
A roadmap to the (vanishing of the) Euler characteristic
We give a diagrammatic summary of the connections between various theorems and conjectures about the vanishing of the Euler characteristic.
math.GT↗
Exponential growth rates in hyperbolic groups (after Koji Fujiwara and Zlil Sela)
A classical result of Jørgensen and Thurston shows that the set of volumes of finite volume complete hyperbolic $3$-manifolds is a well-ordered subset of the real numbers of order type~$ω^ω$; moreover, each volume can only be attained by finitely many isometry types of hyperbolic $3$-manifolds. Fujiwara and Sela established a group-theoretic companion of this result: If $Γ$ is a non-elementary hyperbolic group, then the set of exponential growth rates of~$Γ$ is well-ordered, the order type is at least~$ω^ω$, and each growth rate can only be attained by finitely many finite generating sets (up to automorphisms). In this talk, we outline this work of Fujiwara and Sela and discuss related results.
math.GR↗
$L^2$-Betti numbers and computability of reals
We study the computability degree of real numbers arising as $L^2$-Betti numbers or $L^2$-torsion of groups, parametrised over the Turing degree of the word problem.
math.GR↗
A comment on the structure of graded modules over graded principal ideal domains in the context of persistent homology
The literature in persistent homology often refers to a "structure theorem for finitely generated graded modules over a graded principal ideal domain". We clarify the nature of this structure theorem in this context.
math.AC↗
The spectrum of simplicial volume with fixed fundamental group
We study the spectrum of simplicial volume for closed manifolds with fixed fundamental group and relate the gap problem to rationality questions in bounded (co)homology. In particular, we show that in many cases this spectrum has a gap at zero. For such groups, this leads to corresponding gap results for the minimal volume entropy semi-norm and for the minimal volume entropy in dimension 4.
math.GT↗
Bounded cohomology of finitely presented groups: vanishing, non-vanishing, and computability
We provide new computations in bounded cohomology: A group is boundedly acyclic if its bounded cohomology with trivial real coefficients is zero in all positive degrees. We show that there exists a continuum of finitely generated non-amenable boundedly acyclic groups and construct a finitely presented non-amenable boundedly acyclic group. On the other hand, we construct a continuum of finitely generated groups, whose bounded cohomology has uncountable dimension in all degrees greater than or equal to~$2$, and a concrete finitely presented one. Countable non-amenable groups with these two extreme properties were previously known to exist, but these constitute the first finitely generated/finitely presented examples. Finally, we show that various algorithmic problems on bounded cohomology are undecidable.
math.GR↗
Median quasimorphisms on CAT(0) cube complexes and their cup products
Cup products provide a natural approach to access higher bounded cohomology groups. We extend vanishing results on cup products of Brooks quasimorphisms of free groups to cup products of median quasimorphisms, i.e., Brooks-type quasimorphisms of group actions on CAT(0) cube complexes. In particular, we obtain such vanishing results for groups acting on trees and for right-angled Artin groups. Moreover, we outline potential applications of vanishing results for cup products in bounded cohomology.
math.GR↗
Amenable covers and integral foliated simplicial volume
In analogy with ordinary simplicial volume, we show that integral foliated simplicial volume of oriented closed connected aspherical $n$-manifolds that admit an open amenable cover of multiplicity at most $n$ is zero. This implies that the fundamental groups of such manifolds have fixed price and are cheap as well as reproves some statements about homology growth.
math.GT↗
On the simplicial volume and the Euler characteristic of (aspherical) manifolds
A well-known question by Gromov asks whether the vanishing of the simplicial volume of oriented closed connected aspherical manifolds implies the vanishing of the Euler characteristic. We study various versions of Gromov's question and collect strategies towards affirmative answers and strategies towards negative answers to this problem. Moreover, we put Gromov's question into context with other open problems in low- and high-dimensional topology. A special emphasis is put on a comparative analysis of the additivity properties of the simplicial volume and the Euler characteristic for manifolds with boundary. We explain that the simplicial volume defines a symmetric monoidal functor (TQFT) on the amenable cobordism category, but not on the whole cobordism category. In addition, using known computations of simplicial volumes, we conclude that the fundamental group of the 4-dimensional amenable cobordism category is not finitely generated. We also consider new variations of Gromov's question. Specifically, we show that counterexamples exist among aspherical spaces that are only homology equivalent to oriented closed connected manifolds.
math.AT↗
Bounded cohomology and binate groups
A group is boundedly acyclic if its bounded cohomology with trivial real coefficients vanishes in all positive degrees. Amenable groups are boundedly acyclic, while the first non-amenable examples were the group of compactly supported homeomorphisms of $\mathbb{R}^n$ (Matsumoto--Morita) and mitotic groups (Löh). We prove that binate (alias pseudo-mitotic) groups are boundedly acyclic, which provides a unifying approach to the aforementioned results. Moreover, we show that binate groups are universally boundedly acyclic. We obtain several new examples of boundedly acyclic groups as well as computations of the bounded cohomology of certain groups acting on the circle. In particular, we discuss how these results suggest that the bounded cohomology of the Thompson groups $F$, $T$, and $V$ is as simple as possible.
math.GR↗
Bounded acyclicity and relative simplicial volume
We provide new vanishing and glueing results for relative simplicial volume, following up on two current themes in bounded cohomology: The passage from amenable groups to boundedly acyclic groups and the use of equivariant topology. More precisely, we consider equivariant nerve pairs and relative classifying spaces for families of subgroups. Typically, we apply this to uniformly boundedly acyclic families of subgroups. Our methods also lead to vanishing results for $\ell^2$-Betti numbers of aspherical CW-pairs with small relative amenable category and to a relative version of a result by Dranishnikov and Rudyak concerning mapping degrees and the inheritance of freeness of fundamental groups.
math.AT↗
Topological volumes of fibrations: A note on open covers
We establish a straightforward estimate for the number of open sets with fundamental group constraints needed to cover the total space of fibrations. This leads to vanishing results for simplicial volume and minimal volume entropy, e.g., for certain mapping tori.
math.GT↗