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Tangle free permutations and the Putman-Wieland property of Random covers

Adam Klukowski, Vladimir Marković

Abstract

Let $Σ^p_g$ denote a surface of genus $g$ and with $p$ punctures. Our main result is that the fraction of degree $n$ covers of $Σ^p_g$ which have the Putman-Wieland property tends to $1$ as $n\to \infty$. In addition, we show that the monodromy of a random cover of $Σ^p_g$ is asymptotically almost surely tangle free.

Tangle free permutations and the Putman-Wieland property of Random covers

Abstract

Let denote a surface of genus and with punctures. Our main result is that the fraction of degree covers of which have the Putman-Wieland property tends to as . In addition, we show that the monodromy of a random cover of is asymptotically almost surely tangle free.
Paper Structure (28 sections, 21 theorems, 59 equations, 3 figures)

This paper contains 28 sections, 21 theorems, 59 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The Putman-Wieland conjecture
  3. A random subgroup has the Putman-Wieland property
  4. Random permutations are tangle free
  5. Organisation of the paper
  6. Spectral gap and the Putman-Wieland property
  7. Pointed holomorphic covers
  8. A random cover retains the spectral gap
  9. The $L$-horoball property
  10. The $L$-horoball property and the Cheeger constant
  11. The $L$-horoball property and the spectral gap
  12. A random cover has the $L$-horoball property
  13. Proof of Theorem \ref{['thm-main']}
  14. Finite covers with labelled fibres
  15. Labelled fibres and the Symmetric group
  16. ...and 13 more sections

Key Result

Theorem 1.4

For each $p\ge 0$ there exists $g_0 \in \mathbb{N}$ such that when $g\ge g_0$.

Figures (3)

  • Figure 1: The lollipops $b_1$ (blue), and $b_2$ (red) represent elements of the fundamental group of $\pi_1(X,x)$
  • Figure 2: Both permutations $\phi(w_1), \phi(w_2)$ fix $1$. Vertices represent the orbits of the common fixed point 1 under $\phi(w_1)$ and $\phi(w_2)$, and are labelled according the vertex labelling $f$.
  • Figure 3: The graph $G'$ from Section 7.1. The pairs of vertices with the labels 5 and 6 were identified by the relation $\sim_V$. The graph $G_1$ is then obtained from $G'$ by replacing the two blue edges of label 1 between the vertices 1 and 5 with a single edge. Then $\chi(G_1)=-2$.

Theorems & Definitions (61)

  • Definition 1.1
  • Conjecture 1.2
  • Definition 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Definition 1.6
  • Theorem 1.7
  • Remark 1
  • Remark 2
  • Definition 2.1
  • ...and 51 more