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On the integral simplicial volume of cyclic covers of mapping tori

Federica Bertolotti, Ervin Hadziosmanovic

TL;DR

The paper analyzes the asymptotics of the integral simplicial volume for cyclic covers of torus-fibered mapping tori, introducing the integral filling volume $FV_{\mathbb{Z}}$ as the limiting growth rate of mapping-torus covers. It establishes a tight link between this growth and the monodromy via the spectral radius $\rho(f_*)$ on $H_1(T^n;\mathbb{R})$, giving a universal bound $\frac{1}{K_n}\log\rho(f_*) \le \lim_{k\to\infty} \frac{\lVert (T^n)_{f^k}\rVert_{\mathbb{Z}}}{k} \le K_n\log\rho(f_*)$ with a dimension-two refinement. The upper bound in dimension two is achieved by constructing efficient fillings from parallelogram-like and rectangle-like cycles, culminating in a key bound that ties $\lVert f^k_*(c_n)-c_n \rVert_{\text{fill},\mathbb{Z}}$ to $\log\rho(A)$ via the Gelfand formula. The lower bound leverages torsion growth in first homology of mapping tori and Smith normal form to relate growth to eigenvalues of the monodromy, yielding a general inequality that applies beyond torus fibers. Applications connect $FV_{\mathbb{Z}}$ to topological entropy, provide new non-equivalences between $\Delta$-complexity and integral simplicial volume, and demonstrate the positive filling volume for Anosov diffeomorphisms on infranilmanifolds, enriching the understanding of how algebraic dynamics governs topological complexity.

Abstract

In this paper, we investigate the asymptotic behavior of the integral simplicial volume of cyclic covers of manifolds that fiber over the circle with fiber given by an $n$-dimensional torus. By studying the integral filling volume -- an invariant introduced by Frigerio and the first author -- for the monodromy, we establish both lower and upper bounds for the limit of the integral simplicial volume of these covers, normalized by the degree of the covering. These bounds are expressed in terms of the action of the monodromy on the real homology of the fiber. As applications, we establish a close connection between the topological entropy and the integral filling volume of self-homeomorphisms of $n$-dimensional tori, we find new examples for which the Delta-complexity and the integral simplicial volume are not equivalent, and we prove the nonvanishing of the filling volume for Anosov self-diffeomorphisms of infranilmanifolds.

On the integral simplicial volume of cyclic covers of mapping tori

TL;DR

The paper analyzes the asymptotics of the integral simplicial volume for cyclic covers of torus-fibered mapping tori, introducing the integral filling volume as the limiting growth rate of mapping-torus covers. It establishes a tight link between this growth and the monodromy via the spectral radius on , giving a universal bound with a dimension-two refinement. The upper bound in dimension two is achieved by constructing efficient fillings from parallelogram-like and rectangle-like cycles, culminating in a key bound that ties to via the Gelfand formula. The lower bound leverages torsion growth in first homology of mapping tori and Smith normal form to relate growth to eigenvalues of the monodromy, yielding a general inequality that applies beyond torus fibers. Applications connect to topological entropy, provide new non-equivalences between -complexity and integral simplicial volume, and demonstrate the positive filling volume for Anosov diffeomorphisms on infranilmanifolds, enriching the understanding of how algebraic dynamics governs topological complexity.

Abstract

In this paper, we investigate the asymptotic behavior of the integral simplicial volume of cyclic covers of manifolds that fiber over the circle with fiber given by an -dimensional torus. By studying the integral filling volume -- an invariant introduced by Frigerio and the first author -- for the monodromy, we establish both lower and upper bounds for the limit of the integral simplicial volume of these covers, normalized by the degree of the covering. These bounds are expressed in terms of the action of the monodromy on the real homology of the fiber. As applications, we establish a close connection between the topological entropy and the integral filling volume of self-homeomorphisms of -dimensional tori, we find new examples for which the Delta-complexity and the integral simplicial volume are not equivalent, and we prove the nonvanishing of the filling volume for Anosov self-diffeomorphisms of infranilmanifolds.

Paper Structure

This paper contains 13 sections, 24 theorems, 148 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Integral filling volume and integral simplicial volume
  4. Straight simplices on the torus
  5. Prism operator on topological spaces
  6. Matrices and self-homotopy equivalences of tori
  7. Lower bound
  8. Upper bound in dimension two
  9. Parallelogram and Rectangle-like cycles
  10. Splitting of parallelogram-like cycles
  11. Slim parallelogram-like cycles
  12. Rectangle-like cycles
  13. Proof of \ref{['metteresuS1']}

Key Result

Theorem 1

Fix $n\in {\mathbb{N}}$. There exists a constant $K_n>0$ such that the following holds: if $f\colon T^{n}\to T^{n}$ is an orientation-preserving homeomorphism on the $n$-torus and $f_*\colon {H_{{1}}(T^{n}; {\mathbb{Z}})}\to {H_{{1}}(T^{n}; {\mathbb{Z}})}$ denotes the induced map in homology, then where $\rho(f_*)$ denotes the spectral radius of the linear map $f_*$.

Figures (4)

  • Figure 1: Representations of the canonical lifts in ${\mathbb{R}}^2$ of the parallelogram-like cycle $P^{2}_{2}\left(v,w\right)$ in $T^{2}$ (on the left) and of a rectangle-like cycle (on the right).
  • Figure 2: On the left, a ($3$-dimensional) representation of a lift in ${\mathbb{R}}^3$ of the $3$-chain $Q$ of \ref{['lemma-splitting-vector']}. The boundary of this lift is made up of: the two forward parallelograms, representing lifts of $P^{2}_{2}\left(v_1,v_2'\right)$ and $P^{2}_{2}\left(v_1,v_2"\right)$, the backward parallelogram, representing a lift of $P^{2}_{2}\left(v_1,v_2'+v_2"\right)$, and the two triangles on the left and on the right, given by $R(b_1^+)$ and $R(b_1^-)$, which simplify when projecting on $T^{2}$. The red segment is the support of $P^{2}_{2}\left(v_1,\mathbf 0\right)$. On the right, it is represented the same picture in case of rectangle-like cycles (\ref{['step-splitting-rect']}).
  • Figure 3: To prove \ref{['lemma-rect-with-size1-hdim']} for $n=2$, we apply the slidings of \ref{['step-slide-paral-along-a-side']} four times as shown in the picture, where at each step the black parallelogram is the starting one, and the red parallelogram is the one we obtain after the application of \ref{['step-slide-paral-along-a-side']}. Notice that in the case $a_1=1$, we already have a rectangle like cycle at the third application of \ref{['step-slide-paral-along-a-side']}.
  • Figure :

Theorems & Definitions (60)

  • Theorem 1
  • Theorem 2
  • Corollary 3
  • proof
  • Corollary 4
  • proof
  • Definition 5: smale1967differentiable
  • Definition 6
  • Corollary 7
  • Definition 2.2: bertolotti2024length
  • ...and 50 more