[Paper Review] Volume entropy and the Gromov boundary of flat surfaces

Klaus Dankwart|arXiv (Cornell University)|Jan 10, 2011

Mathematical Dynamics and Fractals4 references3 citations

TL;DR

This paper establishes that for closed flat surfaces of genus $g \geq 2$ and area 1, divergence in the moduli space is equivalent to both the volume entropy and the Hausdorff dimension of the Gromov boundary of the universal cover tending to infinity. It further provides bounds on the entropy of branched coverings in terms of the base surface's entropy, the minimal distance between branch points, and a combinatorial complexity measure, with asymptotically sharp examples constructed.

ABSTRACT

We consider the volume entropy of closed flat surfaces of genus $g\geq 2$ and area 1. We show that a sequence of flat surfaces diverges in the moduli space if and only if the volume entropy converges to infinity. Equivalently the Hausdorff dimension of the Gromov boundary of the isometric universal cover tends to infinity. Moreover, we estimate the entropy of a locally isometric branched covering of a flat surface by the entropy of base surface and the geometry of the covering map.

Motivation & Objective

• To characterize when a sequence of flat surfaces diverges in moduli space using geometric invariants.
• To establish the equivalence between divergent sequences in moduli space and divergent volume entropy or Gromov boundary dimension.
• To estimate the volume entropy of flat branched coverings in terms of geometric and combinatorial data of the covering map.
• To construct examples showing that the derived entropy bounds are asymptotically sharp.

Proposed method

• Uses the Švarc-Milnor lemma to relate the universal cover of a flat surface to the Poincaré disc via quasi-isometry.
• Applies Gromov's theory of $\delta$-hyperbolic spaces to define the Gromov boundary and its Hausdorff dimension.
• Defines volume entropy as the growth rate of the number of deck transformations within a given distance from a basepoint.
• Establishes a key relation: Hausdorff dimension of the Gromov boundary equals $e(X)/\log(\xi)$, where $\xi = 2^{1/(2\delta_{\inf})}$.
• Constructs a mapping $\Phi$ from loops in the covering surface to loops in a model 3-sheeted covering, using branch point data and geodesic concatenation.
• Derives entropy bounds via counting arguments: $e(T) \leq (e(S)+1)\left(a(S) + \frac{C \log(\lambda(T))}{l_b(T)}\right)$, with $\lambda(T)$ measuring combinatorial complexity and $l_b(T)$ the minimal branch point distance.

Experimental results

Research questions

• RQ1 When does a sequence of flat surfaces diverge in the moduli space $\mathcal{Q}_g$?
• RQ2 How are the volume entropy and the Hausdorff dimension of the Gromov boundary related to the geometry of the moduli space?
• RQ3 What bounds can be placed on the volume entropy of a flat branched covering in terms of the base surface and covering geometry?
• RQ4 Are the derived entropy bounds asymptotically sharp?

Key findings

• A sequence of flat surfaces $S_i \in \mathcal{Q}_g$ diverges in moduli space if and only if the volume entropy of $S_i$ tends to infinity.
• Equivalently, the Hausdorff dimension of the Gromov boundary of the universal cover of $S_i$ tends to infinity as $i \to \infty$.
• For a flat branched covering $\pi: T \to S$, the volume entropy satisfies $e(T) \leq (e(S)+1)\left(a(S) + \frac{C \log(\lambda(T))}{l_b(T)}\right)$ for some constant $C > 0$, where $\lambda(T)$ combines sheet number and branch point count, and $l_b(T)$ is the minimal distance between branch points.
• The same entropy bound holds for the Hausdorff dimension of the Gromov boundary.
• The paper constructs a family of examples demonstrating that the entropy bounds are asymptotically sharp.
• The entropy of a 3-sheeted branched covering $T_0 \to S$ with one ramification point is bounded by $e(T_0) \leq a_1(S)(\log(2n) + e(S))$, which is used as a base case in the general bound.