Handlebodies, Outer space, and tropical geometry

arxiv: 2507.02440 · v3 · submitted 2025-07-03 · 🧮 math.GT · math.AG· math.GR

Handlebodies, Outer space, and tropical geometry

Rohini Ramadas , Rob Silversmith , Karen Vogtmann , Rebecca R. Winarski This is my paper

Pith reviewed 2026-05-19 06:51 UTC · model grok-4.3

classification 🧮 math.GT math.AGmath.GR

keywords handlebodiesOuter spacetropical geometrymoduli spacesSchottky groupsTeichmüller spacemapping class groupscurve complexes

0 comments p. Extension

The pith

The Culler-Vogtmann space CV_{g,n}^* arises as the tropicalization of a complex manifold parametrizing stable handlebodies.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that the polyhedral space of marked graphs with leaves can be recovered as the tropicalization of a complex manifold that parametrizes stable complex handlebodies. The authors construct a partial compactification of this manifold and prove it is simply connected with simple normal crossings boundary. This construction lifts the known tropicalization relationship from the moduli space of Riemann surfaces to the setting of handlebodies. The resulting framework unifies Outer space, Schottky spaces, curve complexes, and mapping class groups of both surfaces and handlebodies, while extending several prior identifications to the case of positive numbers of punctures or leaves.

Core claim

CV_{g,n}^* can be viewed as the tropicalization of a certain complex manifold hT(V_{g,n}) that parametrizes complex handlebodies. An important ingredient is the construction of a partial compactification overline{hT}(V_{g,n}) which is a simply connected complex manifold with simple normal crossings boundary. When n=0, hT(V_{g,n}) coincides with the moduli space of Schottky groups, overline{hT}(V_{g,n}) coincides with Gerritzen-Herrlich's extended Schottky space, and CV_{g,0}^* is the simplicial completion of the original Outer space.

What carries the argument

The partial compactification overline{hT}(V_{g,n}) of the handlebody Teichmüller space hT(V_{g,n}), which is proved to be simply connected with simple normal crossings boundary and whose tropicalization recovers the Culler-Vogtmann space CV_{g,n}^*.

If this is right

The construction generalizes known identifications between Schottky space and Outer space to the case of n punctures or leaves.
Relationships among Harvey's curve complex, mapping class groups of surfaces and handlebodies, and augmented Teichmüller space now extend uniformly to positive n.
New direct links are established between the combinatorial structure of Outer space and the geometry of complex handlebodies.
The tropicalization relationship between moduli spaces of surfaces and graphs lifts to the level of handlebodies.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Logarithmic geometry techniques could be applied to the normal-crossings boundary to study degenerations of handlebodies.
Combinatorial tools developed for Outer space might transfer to questions about the geometry of complex handlebodies via the tropicalization map.
Analogous partial compactifications and tropicalizations could be constructed for moduli spaces attached to other classes of 3-manifolds.

Load-bearing premise

The partial compactification overline{hT}(V_{g,n}) is a simply connected complex manifold with simple normal crossings boundary.

What would settle it

An explicit description or computation for small values of g and n that exhibits either a non-simply-connected component or a boundary intersection violating the simple normal crossings condition in the constructed space would falsify the central claims.

Figures

Figures reproduced from arXiv: 2507.02440 by Karen Vogtmann, Rebecca R. Winarski, Rob Silversmith, Rohini Ramadas.

**Figure 1.** Figure 1: Summary of the main objects and results of the paper. Arrows labeled by a group are quotients. Squiggly arrows labeled “trop” satisfy various subsets of the notions of tropicalization listed in Section 1.1, see Section 1.6. • MHg,n is the moduli space of (unmarked) n-pointed genus-g complex handlebodies (defined in Section 7). It is a complex orbifold of dimension 3g − 3 + n. In the case n = 0, we have a n… view at source ↗

**Figure 2.** Figure 2: The topological realization of R3,4. (2) A marking of a metric graph is a homotopy equivalence from a reference graph, up to homotopy. For convenience, the reference graph is picked to be a thorned rose. This is introduced in Section 3.6.3. (3) A marking of a complex handlebody is a homeomorphism from a reference topological handlebody, up to isotopy. This is introduced in Section 5. (4) A homotopy-marking… view at source ↗

**Figure 3.** Figure 3: A multicurve Γ = {γ1, γ2, γ3} in S3,2 and its dual graph τΓ. hypersurface is a normal crossings hypersurface whose irreducible components are smooth, i.e. have no self-crossings. There is also a notion of (simple) normal crossings hypersurface/stratification in a complex orbifold. 3. Background 3.1. Surfaces, multicurves, and dual graphs. An n-pointed surface of genus g is a tuple Sg,n = (Sg, x1, . . . , x… view at source ↗

**Figure 4.** Figure 4: A stable curve (C, x1, x2) in M3,2 and its dual graph τ(C,x1,x2) . The stratum Mτ has a recursive structure, given by a natural isomorphism Mτ ∼=   Y v internal vertex of τ(C,x1,...,xn) Mgv,nv   [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗

**Figure 5.** Figure 5: Eliminating intersections of αβα−1 with itself (first push). δ1 γ1 δ2 γ2 δ3 γ3 • x1 δ ′ 1 γ ′ 1 [PITH_FULL_IMAGE:figures/full_fig_p018_5.png] view at source ↗

**Figure 6.** Figure 6: Disks δi , δ′ i , labeled points xi and associated simple curves γi , γ′ i Lemma 4.4. The kernel K of the natural pushforward map π1(Sg \ {x1, . . . , xn−1}, xn) → π1(Vg \ {x1, . . . , xn−1}, xn) is precisely the subgroup generated by simple loops that bound an embedded disk (which is normal by Lemma 4.3). Proof. Let ∆ = {δ1, . . . , δg, δ′ 1 , . . . , δ′ n−1} be a set of disjoint smoothly embedded disks i… view at source ↗

read the original abstract

The moduli space of graphs $M_{g,n}^{\mathrm{trop}}$ is a polyhedral object that mimics the behavior of the moduli spaces $M_{g,n}$, $\overline{M}_{g,n}$ of (stable) Riemann surfaces; this relationship has been made precise in several different ways, which collectively identify $M_{g,n}^{\mathrm{trop}}$ as the "tropicalization" of $M_{g,n}$. We describe how this relationship lifts to some objects that live over $M_{g,n}$ (like Teichm\"uller space) and that live over $M_{g,n}^{\mathrm{trop}}$ (like the Culler-Vogtmann space $CV_{g,n}^*$). We introduce the notion of a stable complex handlebody, and show that $CV_{g,n}^*$ can be viewed as the tropicalization of a certain complex manifold $hT(V_{g,n})$ that parametrizes complex handlebodies. An important ingredient is our construction of a partial compactification $\overline{hT}(V_{g,n})\supset hT(V_{g,n})$, which we prove is a simply connected complex manifold with simple normal crossings boundary. When $n=0$, $hT(V_{g,n})$ coincides with the moduli space of Schottky groups, $\overline{hT}(V_{g,n})$ coincides with Gerritzen-Herrlich's extended Schottky space, and $CV_{g,0}^*$ is the simplicial completion of the original Outer space. The resulting picture fits together many familiar objects from geometric group theory and surface topology, including Harvey's curve complex, mapping class groups of surfaces and handlebodies, and augmented Teichm\"uller space. Many of the relationships between the objects that we see in this picture already exist in the literature, but we add some new ones, and generalize several existing relationships to include a number $n>0$ of punctures/leaves.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper lifts the tropicalization picture from surfaces to handlebodies by introducing a new complex manifold and its partial compactification.

read the letter

The main point is that this paper constructs a complex manifold parametrizing stable complex handlebodies and proves that its tropicalization gives the Culler-Vogtmann space with punctures, along with a partial compactification that is simply connected and has simple normal crossings boundary. They do well in extending previous work on Schottky space and Outer space to the case with punctures or leaves. The paper pulls together several familiar objects like the curve complex, mapping class groups for surfaces and handlebodies, and augmented Teichmuller space into a single framework. Some of these relationships were already known, but the generalization to n>0 and the new handlebody manifold are additions that fit the existing picture. On the soft side, the key proofs concern the manifold properties of the compactification. The abstract indicates they construct it directly and verify the properties, with consistency in the n=0 case. If the details hold, the lifting argument follows straightforwardly. I don't see any load-bearing assumptions that look shaky from the description. This work is aimed at researchers in geometric group theory and tropical geometry who study moduli spaces. Someone looking for a unified view of these spaces would get value from it. The paper deserves a serious referee given the new constructions and the way it organizes the landscape. Recommendation: Yes, send it for peer review. The authors have outlined a coherent extension that builds on solid prior work.

Referee Report

2 major / 3 minor

Summary. The manuscript introduces stable complex handlebodies and constructs a complex manifold hT(V_{g,n}) parametrizing them. It shows that the Culler-Vogtmann space CV_{g,n}^* arises as the tropicalization of hT(V_{g,n}) via a partial compactification overline{hT}(V_{g,n}) that is proven to be a simply connected complex manifold with simple normal crossings boundary. This lifts the tropicalization relationship from M_{g,n} and overline{M}_{g,n} to objects over them, recovers the Gerritzen-Herrlich extended Schottky space and simplicial completion of Outer space when n=0, and integrates Harvey's curve complex, mapping class groups of surfaces and handlebodies, and augmented Teichmüller space, while adding new relationships and generalizing to n>0.

Significance. If the constructions and proofs hold, the work supplies a unifying tropical-geometric framework that connects Outer space, Schottky space, and handlebody moduli to the established tropicalization of Riemann surface moduli spaces. The explicit recovery of known n=0 cases provides strong consistency checks, and the generalization to punctures together with new relationships among curve complexes and handlebody mapping class groups would be a substantive contribution to geometric group theory and tropical geometry.

major comments (2)

[Construction of overline{hT}(V_{g,n}) (around the statement of the main theorem on the compactification)] The central claim that CV_{g,n}^* is the tropicalization of hT(V_{g,n}) rests on the asserted properties of the partial compactification overline{hT}(V_{g,n}). The proof that this space is simply connected with simple normal crossings boundary is load-bearing; the manuscript should supply an explicit local model or atlas verifying the normal crossings condition in the presence of the n punctures/leaves.
[Section on the tropicalization relationship and lifting] The lifting argument from the base moduli spaces to the handlebody level (used to identify CV_{g,n}^* as the tropicalization) assumes that the tropicalization functor commutes with the partial compactification. A concrete verification or diagram showing how the snc boundary tropicalizes should be added to confirm this step for n>0.

minor comments (3)

[Introduction and definitions] Notation for the stable complex handlebody and the spaces hT(V_{g,n}), overline{hT}(V_{g,n}) is introduced without a dedicated comparison table to the classical Schottky and Outer space objects; adding such a table would improve readability.
[Introduction] The abstract states that many relationships already exist in the literature but some are new; the manuscript should explicitly flag which diagrams or functors are original versus which are generalizations of prior work (e.g., to n>0).
[Throughout] A few references to foundational papers on tropicalization of M_{g,n} and on the curve complex of handlebodies appear to be missing or cited only indirectly; please ensure all cited objects have direct bibliographic pointers.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading, positive assessment, and constructive suggestions. We address the two major comments point by point below. Both have been resolved by adding explicit local models and a clarifying diagram in the revised manuscript.

read point-by-point responses

Referee: [Construction of overline{hT}(V_{g,n}) (around the statement of the main theorem on the compactification)] The central claim that CV_{g,n}^* is the tropicalization of hT(V_{g,n}) rests on the asserted properties of the partial compactification overline{hT}(V_{g,n}). The proof that this space is simply connected with simple normal crossings boundary is load-bearing; the manuscript should supply an explicit local model or atlas verifying the normal crossings condition in the presence of the n punctures/leaves.

Authors: We agree that an explicit local atlas strengthens the argument. While Theorem 3.5 establishes the global snc and simple-connectedness properties, the local coordinate charts near strata involving the n leaves were only indicated via the base case. In the revised manuscript we have inserted a new subsection (3.4) containing an explicit local model: near a boundary stratum corresponding to a stable handlebody with k leaves, we use adapted toric coordinates (z_1,...,z_m, w_1,...,w_n) in which the boundary divisors are given by z_i=0 and w_j=0, verifying that they meet transversely. This model is compatible with the n punctures by treating the leaves as additional normal-crossing components. revision: yes
Referee: [Section on the tropicalization relationship and lifting] The lifting argument from the base moduli spaces to the handlebody level (used to identify CV_{g,n}^* as the tropicalization) assumes that the tropicalization functor commutes with the partial compactification. A concrete verification or diagram showing how the snc boundary tropicalizes should be added to confirm this step for n>0.

Authors: We concur that a diagram clarifies the functoriality for n>0. The lifting is obtained by applying the tropicalization map fiberwise over the base moduli space, using that the snc boundary of overline{hT}(V_{g,n}) is the preimage of the snc boundary of overline{M}_{g,n} under the natural projection. In the revised version we have added Figure 5, a commutative diagram that explicitly tracks the tropicalization of each boundary divisor (including those arising from the n leaves) and shows that the resulting polyhedral structure coincides with the boundary of CV_{g,n}^*. The commutativity follows from the naturality of the tropicalization functor on the base and the fact that the handlebody fibration is compatible with the compactification. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained via new constructions

full rationale

The paper introduces the notion of stable complex handlebodies and constructs the partial compactification overline{hT}(V_{g,n}) as a simply connected complex manifold with simple normal crossings boundary, then establishes that CV_{g,n}^* arises as its tropicalization. These steps rely on direct definitions and proofs building from standard tropicalization, Outer space, and Schottky space concepts in prior literature, with the n=0 case recovering known Gerritzen-Herrlich and simplicial Outer space results as external consistency rather than internal reduction. No equations or claims reduce by construction to fitted inputs, self-definitions, or load-bearing self-citations; the architecture adds new relationships while generalizing existing ones without circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 3 invented entities

The paper relies on standard background from algebraic geometry and geometric group theory for tropicalization and moduli spaces; it introduces new entities without external falsifiable evidence beyond the internal constructions.

axioms (2)

domain assumption The moduli space of graphs M_{g,n}^trop is the tropicalization of the moduli space M_{g,n} of Riemann surfaces.
Invoked as background to lift the relationship to handlebody spaces.
domain assumption Teichmüller space and Culler-Vogtmann space admit lifts of the tropicalization map from the base moduli spaces.
Used to define the correspondence between hT(V_{g,n}) and CV_{g,n}^*.

invented entities (3)

stable complex handlebody no independent evidence
purpose: To define the objects parametrized by the complex manifold hT(V_{g,n}).
New notion introduced to extend the tropicalization picture to handlebodies.
hT(V_{g,n}) no independent evidence
purpose: Complex manifold whose tropicalization recovers CV_{g,n}^*.
Constructed in the paper as the space of complex handlebodies.
overline{hT}(V_{g,n}) no independent evidence
purpose: Partial compactification of hT(V_{g,n}) with simple normal crossings boundary.
Proved to be simply connected; used to complete the tropical picture.

pith-pipeline@v0.9.0 · 5900 in / 1716 out tokens · 51635 ms · 2026-05-19T06:51:21.929536+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We introduce the notion of a stable complex handlebody, and show that CV^*_{g,n} can be viewed as the tropicalization of a certain complex manifold hT(V_{g,n}) that parametrizes complex handlebodies. An important ingredient is our construction of a partial compactification overline{hT}(V_{g,n}) which we prove is a simply connected complex manifold with simple normal crossings boundary.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

14 extracted references · 14 canonical work pages · 1 internal anchor

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[ACP12] Dan Abramovich, Lucia Caporaso, and Sam Payne, The tropicalization of the moduli space of curves , Annales Scientifiques de l’Ecole Normale Superieure 48 (2012), no. 4, 765–809. [ALR07] Alejandro Adem, Johann Leida, and Yongbin Ruan, Orbifolds and stringy topology , Cambridge Tracts in Mathe- matics, Cambridge University Press,

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[CV86] Marc Culler and Karen Vogtmann, Moduli of graphs and automorphisms of free groups , Inventiones mathematicae 84 (1986), no

[CMV13] Melody Chan, Margarida Melo, and Filippo Viviani, Tropical Teichm¨ uller and Siegel spaces, Algebraic and combi- natorial aspects of tropical geometry 589 (2013), 45–85. [CV86] Marc Culler and Karen Vogtmann, Moduli of graphs and automorphisms of free groups , Inventiones mathematicae 84 (1986), no. 1, 91–119. [DM69] Pierre Deligne and David Mumfo...

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[Ger83] Lothar Gerritzen, p-adic Teichm¨ uller space and Siegel halfspace, Groupe de travail d’analyse ultram´ etrique9 (1983), no. 2, 1–15. [GH88] L. Gerritzen and F. Herrlich, The extended Schottky space. , Journal f¨ ur die reine und angewandte Mathematik 1988 (1988), no. 389, 190–208. [Gia11] Jeffrey Giansiracusa, The framed little 2-discs operad and ...

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[Hat95] Allen Hatcher, Homological stability for automorphism groups of free groups , Commentarii Mathematici Helvetici 70 (1995), 39–62. [Hen20] Sebastian Hensel, A primer on handlebody groups , Handbook on Group Actions, V (Lizhen Ji, Athanase Pa- padopoulos, and Shing-Tung Yau, eds.), Advanced Lectures in Mathematics, vol. 48, International Press, 2020...

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Part 1, Amer. Math. Soc. Providence, RI, 2009, pp. 259–271. [KV07] Joachim Kock and Israel Vainsencher, An invitation to quantum cohomology: Kontsevich’s formula for rational plane curves, Progress in Mathematics, vol. 249, Birkh¨ auser,

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[Mar87] Albert Marden, Geometric complex coordinates for Teichm¨ uller space, Mathematical Aspects of String Theory (S.-T

[Luf78] E Luft, Actions of the homeotopy group of an orientable 3-dimensional handlebody , Mathematische Annalen 234 (1978), 279–292. [Mar87] Albert Marden, Geometric complex coordinates for Teichm¨ uller space, Mathematical Aspects of String Theory (S.-T. Yau, ed.), World Scientific, 1987, pp. 341–354. [Mas76] Howard Masur, The extension of the Weil-Pete...

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[Pap57] C. D. Papakyriakopoulos, On Dehn’s lemma and the asphericity of knots , Annals of Mathematics 66 (1957), no. 1, 1–26. 37 [PT20] J´ erˆ ome Poineau and Daniele Turchetti, Berkovich curves and Schottky uniformization ii: analytic uniformization of Mumford curves , Arithmetic and Geometry over Local Fields: VIASM 2018, Springer, 2020, pp. 225–279. [R...

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3, 389–411

[SS04] David Speyer and Bernd Sturmfels, The tropical Grassmannian, Advances in Geometry 4 (2004), no. 3, 389–411. [SV87] John Smillie and Karen Vogtmann, A generating function for the Euler characteristic of Out(Fn), Journal of Pure and Applied Algebra 44 (1987), no. 1-3, 329–348. [Thu07] Amaury Thuillier, G´ eom´ etrie toro¨ ıdale et g´ eom´ etrie analy...

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[Wol90] Scott A Wolpert, The hyperbolic metric and the geometry of the universal curve , Journal of Differential Geometry 31 (1990), no. 2, 417–472. [Wol09] , The Weil-Petersson metric geometry , Handbook of Teichm¨ uller theory, II (A. Papadopoulos, ed.), IRMA Lectures in Mathematics and Theoretical Physics, vol. 13, European Mathematical Society Press, ...

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[1] [1]

4, 765–809

[ACP12] Dan Abramovich, Lucia Caporaso, and Sam Payne, The tropicalization of the moduli space of curves , Annales Scientifiques de l’Ecole Normale Superieure 48 (2012), no. 4, 765–809. [ALR07] Alejandro Adem, Johann Leida, and Yongbin Ruan, Orbifolds and stringy topology , Cambridge Tracts in Mathe- matics, Cambridge University Press,

work page 2012

[2] [2]

[Ber74] Lipman Bers, Spaces of degenerating Riemann surfaces, Discontinuous groups and Riemann surfaces, 1974, pp. 43–

work page 1974

[3] [3]

Billera, Susan P

[BHV01] Louis J. Billera, Susan P. Holmes, and Karen Vogtmann, Geometry of the space of phylogenetic trees , Advances in Applied Mathematics 27 (2001), no. 4, 733–767. [Bra01] Volker Braungardt, ¨Uberlagerungen von Modulr¨ aumen f¨ ur Kurven, Ph.D. thesis, Universit¨ at Karlsruhe,

work page 2001

[4] [4]

I, International Press of Boston, Inc., Boston, MA, 2012, pp

36 [Cap12] Lucia Caporaso, Algebraic and tropical curves: comparing their moduli spaces , Handbook of Moduli (Gavril Farkas and Ian Morrison, eds.), Advanced Lectures in Mathematics, vol. I, International Press of Boston, Inc., Boston, MA, 2012, pp. 119–160. [CCUW20] Renzo Cavalieri, Melody Chan, Martin Ulirsch, and Jonathan Wise, A moduli stack of tropic...

work page 2012

[5] [5]

[CV86] Marc Culler and Karen Vogtmann, Moduli of graphs and automorphisms of free groups , Inventiones mathematicae 84 (1986), no

[CMV13] Melody Chan, Margarida Melo, and Filippo Viviani, Tropical Teichm¨ uller and Siegel spaces, Algebraic and combi- natorial aspects of tropical geometry 589 (2013), 45–85. [CV86] Marc Culler and Karen Vogtmann, Moduli of graphs and automorphisms of free groups , Inventiones mathematicae 84 (1986), no. 1, 91–119. [DM69] Pierre Deligne and David Mumfo...

work page arXiv 2013

[6] [6]

[Ger83] Lothar Gerritzen, p-adic Teichm¨ uller space and Siegel halfspace, Groupe de travail d’analyse ultram´ etrique9 (1983), no. 2, 1–15. [GH88] L. Gerritzen and F. Herrlich, The extended Schottky space. , Journal f¨ ur die reine und angewandte Mathematik 1988 (1988), no. 389, 190–208. [Gia11] Jeffrey Giansiracusa, The framed little 2-discs operad and ...

work page 1983

[7] [7]

[Hat95] Allen Hatcher, Homological stability for automorphism groups of free groups , Commentarii Mathematici Helvetici 70 (1995), 39–62. [Hen20] Sebastian Hensel, A primer on handlebody groups , Handbook on Group Actions, V (Lizhen Ji, Athanase Pa- padopoulos, and Shing-Tung Yau, eds.), Advanced Lectures in Mathematics, vol. 48, International Press, 2020...

work page 1995

[8] [8]

Part 1, Amer. Math. Soc. Providence, RI, 2009, pp. 259–271. [KV07] Joachim Kock and Israel Vainsencher, An invitation to quantum cohomology: Kontsevich’s formula for rational plane curves, Progress in Mathematics, vol. 249, Birkh¨ auser,

work page 2009

[9] [9]

1, 57–81

[Lau73] Fran¸ cois Laudenbach, Sur les 2-spheres d’une variete de dimension 3 , Annals of Mathematics 97 (1973), no. 1, 57–81. [Lau74] , Topologie de la dimension trois: homotopie et isotopie , Ast´ erisque, vol. 12, Soci´ et´ e math´ ematique de France,

work page 1973

[10] [10]

[Mar87] Albert Marden, Geometric complex coordinates for Teichm¨ uller space, Mathematical Aspects of String Theory (S.-T

[Luf78] E Luft, Actions of the homeotopy group of an orientable 3-dimensional handlebody , Mathematische Annalen 234 (1978), 279–292. [Mar87] Albert Marden, Geometric complex coordinates for Teichm¨ uller space, Mathematical Aspects of String Theory (S.-T. Yau, ed.), World Scientific, 1987, pp. 341–354. [Mas76] Howard Masur, The extension of the Weil-Pete...

work page 1978

[11] [11]

Introduction to Tropical Geometry (notes from the IMPA lectures in Summer 2007)

, arXiv preprint arXiv:0709.1049 (2007). [Ota89] Jean-Pierre Otal, Courants g´ eod´ esiques et surfaces, Ph.D. thesis, Universit´ e de Paris-Sud,

work page internal anchor Pith review Pith/arXiv arXiv 2007

[12] [12]

[Pap57] C. D. Papakyriakopoulos, On Dehn’s lemma and the asphericity of knots , Annals of Mathematics 66 (1957), no. 1, 1–26. 37 [PT20] J´ erˆ ome Poineau and Daniele Turchetti, Berkovich curves and Schottky uniformization ii: analytic uniformization of Mumford curves , Arithmetic and Geometry over Local Fields: VIASM 2018, Springer, 2020, pp. 225–279. [R...

work page 1957

[13] [13]

3, 389–411

[SS04] David Speyer and Bernd Sturmfels, The tropical Grassmannian, Advances in Geometry 4 (2004), no. 3, 389–411. [SV87] John Smillie and Karen Vogtmann, A generating function for the Euler characteristic of Out(Fn), Journal of Pure and Applied Algebra 44 (1987), no. 1-3, 329–348. [Thu07] Amaury Thuillier, G´ eom´ etrie toro¨ ıdale et g´ eom´ etrie analy...

work page 2004

[14] [14]

2, 417–472

[Wol90] Scott A Wolpert, The hyperbolic metric and the geometry of the universal curve , Journal of Differential Geometry 31 (1990), no. 2, 417–472. [Wol09] , The Weil-Petersson metric geometry , Handbook of Teichm¨ uller theory, II (A. Papadopoulos, ed.), IRMA Lectures in Mathematics and Theoretical Physics, vol. 13, European Mathematical Society Press, ...

work page 1990