Handlebodies, Outer space, and tropical geometry

1. Problem Statement and Motivation

The paper addresses the relationship between Geometric Group Theory (specifically Outer Space and automorphisms of free groups), Low-Dimensional Topology (handlebodies and mapping class groups), and Tropical/Non-Archimedean Geometry (moduli spaces of tropical curves).

While the moduli space of Riemann surfaces, $\mathcal{M}_{g,n}$, and its tropicalization, $\mathcal{M}^{\text{trop}}_{g,n}$, are well-understood via the boundary complex of the Deligne-Mumford compactification, the analogous relationship for Outer Space ($CV_{g,n}$) and the moduli space of handlebodies has been less precise.

• The Gap: Outer Space $CV_{g,n}$ (parametrizing marked metric graphs) behaves like a tropical moduli space, but it is not the boundary complex of an algebraic variety. The authors seek to identify a "complex manifold" that lives "over" Outer Space in the same way Teichmüller space lives over $\mathcal{M}_{g,n}$, thereby establishing a rigorous "tropicalization" framework for handlebodies.
• The Goal: To construct a complex manifold $\mathfrak{hT}(V_{g,n})$ (homotopy Teichmüller space) whose boundary complex is the simplicial completion of Outer Space, $CV^*_{g,n}$, and to generalize these relationships to include $n > 0$ punctures (labeled points).

2. Methodology and Key Objects

The authors construct a commutative diagram linking complex geometric objects with their tropical/combinatorial counterparts. The methodology involves:

• Defining Complex Handlebodies:

  • $T(V_{g,n})$: The Teichmüller space of a handlebody, defined as the space of complex structures on a handlebody $V_{g,n}$ with a topological marking. It is shown to be isomorphic to the standard Teichmüller space $T(S_{g,n})$ of the boundary surface.
  • $\mathfrak{hT}(V_{g,n})$: A new space called Homotopy Teichmüller space. It parametrizes "homotopy-marked" complex handlebodies. Crucially, the marking is a homotopy equivalence rather than a homeomorphism. This space is a partial compactification of $T(V_{g,n})$ allowing for "stable" complex handlebodies (handlebodies glued at nodes).
  • $MH_{g,n}$: The moduli space of unmarked complex handlebodies (quotient of $\mathfrak{hT}$ by the homotopy mapping class group).

• Group Theoretic Analysis:

  • The paper analyzes the Mapping Class Group of a Handlebody, $MCG(V_{g,n})$, and its Twist Subgroup, $Tw(V_{g,n})$ (generated by Dehn twists along essential disks).
  • A central technical result is the proof of an exact sequence relating these groups to the Homotopy Mapping Class Group $hMCG(V_{g,n})$ (isomorphic to $Out(F_g) \ltimes F_g^{n-1}$):
    $$1 \to Tw(V_{g,n}) \to MCG(V_{g,n}) \to hMCG(V_{g,n}) \to 1$$
    This generalizes a classical result by Luft (originally for $n=0,1$) to arbitrary $n$.

• Tropicalization via Boundary Complexes:

  • The authors define the "tropicalization" of these spaces as their boundary complexes (the combinatorial structure of the boundary strata).
  • They establish that the boundary complex of $\mathfrak{hT}(V_{g,n})$ is canonically isomorphic to the simplicial completion of Outer Space, $CV^*_{g,n}$.
  • They establish that the boundary complex of $MH_{g,n}$ is canonically isomorphic to the link of the tropical moduli space, $Lk(\mathcal{M}^{\text{trop}}_{g,n})$.

3. Key Contributions and Results

A. The Exact Sequence of Mapping Class Groups (Theorem A)

The authors prove that for any genus $g$ and number of punctures $n$, the sequence
$$1 \to Tw(V_{g,n}) \to MCG(V_{g,n}) \to hMCG(V_{g,n}) \to 1$$
is exact. This identifies the kernel of the map from the topological mapping class group to the homotopy mapping class group precisely as the twist subgroup. This is essential for understanding the quotient structures of the moduli spaces.

B. Construction of Homotopy Teichmüller Space (Theorems B, C, D)

• Definition: $\mathfrak{hT}(V_{g,n})$ is defined as the quotient of the augmented Teichmüller space $T(V_{g,n})$ by the twist subgroup $Tw(V_{g,n})$.
• Complex Structure: $\mathfrak{hT}(V_{g,n})$ is proven to be a simply connected complex manifold of dimension $3g-3+n$.
• Boundary Structure: The boundary $\mathfrak{hT}(V_{g,n}) \setminus \mathfrak{hT}(V_{g,n})$ is a countable union of complex submanifolds meeting at simple normal crossings.
• Relation to Outer Space: The boundary complex of $\mathfrak{hT}(V_{g,n})$ is canonically identified with $CV^*_{g,n}$.

C. Moduli of Handlebodies (Theorem C, Corollary F)

• The moduli space $MH_{g,n}$ (unmarked handlebodies) is shown to be a complex orbifold.
• Its boundary complex is identified with $Lk(\mathcal{M}^{\text{trop}}_{g,n})$, the moduli space of tropical curves with total edge length 1.
• The map $MH_{g,n} \to \mathcal{M}_{g,n}$ (to the moduli space of curves) induces an isomorphism on boundary complexes, despite being infinite-to-one on the interior.

D. Tropicalization Maps (Theorem G)

The authors construct continuous, equivariant surjections analogous to the Chan-Galatius-Payne map:

• $\tilde{\lambda}_{V_{g,n}}: T(V_{g,n}) \to \text{Cone}(D(V_{g,n}))$
• $\lambda^{\mathfrak{hT}}_{g,n}: \mathfrak{hT}(V_{g,n}) \to \text{Cone}(CV^*_{g,n})$
• $\lambda^{MH}_{g,n}: MH_{g,n} \to \mathcal{M}^{\text{trop}}_{g,n}$
  These maps send a complex handlebody to its "tropical limit" (a metric graph) determined by the lengths of the boundaries of essential disks.

E. Simply Connectedness (Theorem 10.1)

A significant topological result is the proof that $\mathfrak{hT}(V_{g,n})$ is simply connected for all $g, n$. This is achieved using the Seifert-van Kampen theorem and an inductive argument on the dimension of the strata, leveraging the contractibility of the underlying augmented Teichmüller space.

4. Significance and Impact

1. Unification of Fields: The paper provides a unified framework connecting:

   • Geometric Group Theory: Outer Space and $Out(F_g)$.
   • Topology: Handlebody mapping class groups and disk complexes.
   • Complex Geometry: Schottky space and Teichmüller theory.
   • Tropical Geometry: Moduli of tropical curves.
     It clarifies that Outer Space is the "tropicalization" of a specific complex manifold ($\mathfrak{hT}$), just as the tropical moduli space is the tropicalization of the Deligne-Mumford compactification.

2. Generalization to $n > 0$: Previous work on Schottky space and Outer Space often focused on the case $n=0$ (no punctures). This paper successfully generalizes these structures to handlebodies with labeled points, which is crucial for applications in phylogenetics and moduli problems with marked points.

3. New Complex Structures: The construction of $\mathfrak{hT}(V_{g,n})$ as a complex manifold with simple normal crossings boundary offers a new geometric object to study. It resolves the issue that $CV^*_{g,n}$ is not the boundary of an algebraic variety by showing it is the boundary of a non-algebraic complex manifold.

4. Schottky Space Interpretation: For $n=0$, $\mathfrak{hT}(V_{g,0})$ coincides with the classical Schottky space, and $MH_{g,0}$ coincides with the unmarked Schottky space. The paper extends the known relationship between Schottky space and Outer Space to the punctured case, providing a "tropicalization" map for Schottky groups with marked orbits.

5. Resolution of "Tropicalization" Ambiguity: The authors carefully distinguish between different levels of "niceness" in tropicalization (algebraic vs. non-algebraic complex manifolds), clarifying why Outer Space cannot be the boundary of an algebraic variety but fits perfectly as the boundary of a complex manifold.

In summary, the paper constructs a rigorous geometric bridge between the combinatorial world of Outer Space and the complex analytic world of handlebodies, proving that the simplicial completion of Outer Space arises naturally as the boundary of a simply connected complex manifold, thereby elevating the status of Outer Space within the landscape of tropical geometry.