Journal of Topology and Analysis

We show that for every finite-volume hyperbolic $3$-manifold $M$ and every prime $p$ we have $dim{H}_1(M;F_p)<168.602\cdot \mathrm{\text{vol}}(M)$. There are slightly stronger estimates if $p=2$ or if $M$ is non-compact. This improves on a result proved by Agol, Leininger and Margalit, which gave the same inequality with a coefficient of $334.08$ in place of $168.602$. It also improves on the analogous result with a coefficient of about $260$, which could have been obtained by combining the arguments due to Agol, Leininger and Margalit with a result due to Böröczky. Our inequality involving homology rank is deduced from a result about the rank of the fundamental group: if $M$ is a finite-volume orientable hyperbolic $3$-manifold such that ${\pi }_1(M)$ is $2$-semifree, then $\mathrm{rank}{\pi }_1(M)<1+\lambda 0\cdot \mathrm{\text{vol}}M$, where $\lambda 0$ is a certain constant less than $167.79$.

We study a pair of $C^{\ast }$-algebras $(A,B)$ by associating a ∗-homomorphism from $A$ to $B$ which allows an approximate left inverse to the ultrapower $C^{\ast }$-algebra of $A$ as a completely positive map of order zero and show that important regularity properties related to the Elliott program pass from $B$ to $A$ in our setting.

We prove equality of analytic and topological $L^2$-torsion associated with an odd-dimensional finite volume hyperbolic manifold and a representation of the fundamental group which extends to the ambient Lie group. This generalizes a previous result due to Lück and Schick. Alternatively, this result can be regarded as the $L^2$-analog of recent work by Müller and Rochon.

In this paper, we provide a characterization of a lower $N$-weighted Ricci curvature bound for $N\in ]-\infty ,1]\cup [n,+\infty ]$ with $𝜀$-range introduced by Lu–Minguzzi–Ohta [Comparison theorems on weighted Finsler manifolds and space-times with $𝜀$-range, Anal. Geom. Metr. Spaces 10(1) (2022) 1–30] in terms of a convexity of entropies over Wasserstein space. We further derive various interpolation inequalities and functional inequalities.

Let $G$ be a finitely presented, residually finite group and let $\delta (G)$ denote the deficiency of $G$. Assume that every subgroup $H$ of finite index in $G$ satisfies $\delta (H)-1=|G:H|(\delta (G)-1)$. We conjecture that $G$ has a two-dimensional finite classifying space $K(G,1)$. This conjecture is motivated by an open question about the deficiency gradient of groups and their $L^2$-Betti numbers. In this note, we relate this conjecture to the relation gap problem for group presentations. We verify the pro-$p$ version of the conjecture, as well as its higher dimensional abstract analogs.

In this paper we study a coarse version of the $K$-theoretic Farrell–Jones conjecture we call coarse or bounded isomorphism conjecture. Using controlled category theory we are able to translate this conjecture for asymptotically faithful covers into a more familiar form. This allows us to prove the conjecture for box spaces of residually finite groups whose Farrell–Jones assembly map with coefficients is an isomorphism.

We prove that finite index subgroups in $S$-arithmetic Chevalley groups are bounded.

Let $M$ be an oriented geometrically finite hyperbolic manifold of infinite volume with dimension $n\ge 3$. For all $k\ge 0$, we provide a lower bound on the $k$th eigenvalue of the Laplace–Beltrami operator of $M$ by a constant and the $k$th eigenvalue of some neighborhood of the thick part of the convex core. As an application, we recover a theorem similar to the one of Burger and Canary which bounds the bottom ${\lambda }_0$ of the spectrum from below by $\frac{c}{\mathrm{\text{vol}}{(C_1(M))}^2}$, where $C_1(M)$ is the $1$-neighborhood of the convex core and $c$ is a constant.

We give a partial classification for the real tight contact structures on solid tori up to equivariant contact isotopy and apply the results to the classification of real tight structures on $S^3$ and real lens spaces $L(p,\pm 1)$. We prove that there is a unique real tight $S^3$ and $ℝP^3$. We show that there is at most one real tight $L(p,\pm 1)$ with respect to one of its two possible real structures. With respect to the other we give lower and upper bounds for the count. To establish lower bounds we explicitly construct real tight manifolds through equivariant contact surgery, real open book decompositions and isolated real algebraic surface singularities. As a by-product we observe the existence of an invariant torus in an $L(p,p-1)$ which cannot be made convex equivariantly.

We analyze all monodromies of genus one fibered knots that possess clean or once-unclean arcs, and use this to determine all manifolds containing genus one fibered knots with generalized crossing changes resulting in another genus one fibered knot, and classify all such generalized crossing changes between two genus one, fibered knots.