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Tommaso Cremaschi

Publications and source records attributed to Tommaso Cremaschi.

17 recordsLinked to original sources
arXiv2026

The Asayama-Matsumoto conjecture and a refined discrepancy bound

Every $n$-vertex plane triangulation admits a polychromatic red--blue vertex coloring with discrepancy at most $n-2\ceil{n/3}\le\floor{n/3}$, resolving a conjecture of Asayama and Matsumoto. The proof follows by combining Loyola et al. 2026 and Kawarabayashi et al. 2026. For $n\ge6$ and $n\not\equiv5\pmod6$, we prove the sharper bound $n-2\ceil{(n+2)/3}$. An exhaustive census of all $9{,}150$ non-isomorphic sphere triangulations with $4\le n\le12$ verifies the bounds and also confirms the sharper estimate at $n=11$, the first order in the remaining open residue class.

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arXiv2026

Monomial web basis for the SL(N) skein algebra of the twice punctured sphere

We give a new proof of a slightly modified version of a result of Queffelec--Rose, by constructing a linear basis for the $\mathrm{SL}(n)$ skein algebra of the twice punctured sphere for any non-zero complex number $q$, excluding finitely many roots of unity of small order. In particular, the skein algebra is a commutative polynomial algebra in $n-1$ generators, where each generator is represented by an explicit $\mathrm{SL}(n)$ web, without crossings, on the surface. This includes the case $q=1$, where the skein algebra is identified with the coordinate ring of the $\mathrm{SL}(n)$ character variety of the twice punctured sphere. The proof of both the spanning and linear independence properties of the basis depends on the so-called $\mathrm{SL}(n)$ quantum trace map, due originally to Bonahon--Wong in the case $n=2$. Two consequences of our method are that the quantum trace map and the so-called splitting map embed the polynomial algebra into the Fock--Goncharov quantum higher Teichmüller space and the Lê--Sikora stated skein algebra, respectively, of the annulus. We end by discussing the relationship with Fock--Goncharov duality.

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arXiv2025

Covers of surfaces

We study the homeomorphism types of certain covers of (always orientable) surfaces, usually of infinite-type. We show that every surface with non-abelian fundamental group is covered by every noncompact surface, we identify the universal abelian covers and the $\mathbb{Z}/n\mathbb{Z}$-homology covers of surfaces, and we show that non-locally finite characteristic covers of surfaces have four possible homeomorphism types.

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arXiv2025

Behaviour of the Schwarzian derivative on long complex projective tubes

The Schwarzian derivative parametrizes the fibres of the space of complex projective structures on a surface as vector bundle over its Teichm\"uller space. We study its behaviour on long complex projective tubes, and get estimates for the pairing of its real part with infinitesimal earthquakes and graftings. As the real part of their Schwarzian coincides with the differential of the renormalized volume we obtain bounds for the variation of renormalized volume under complex earthquake paths, and its asymptotic behaviour under pinching a compressible curve.

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arXiv2024

Density results for the modular group of infinite-type surfaces

In this work we show two results about approximating, with respect to the compact-open topology, mapping classes on surfaces of infinite-type by quasi-conformal maps, in particular we are interested in density results. The first result is that given any infinite-type surface $S$ there exists a hyperbolic structure $X$ on $S$ such that $\text{PMCG}(S)\subseteq \overline{\text{Mod}(X)}$, for $\text{Mod}(X)$ the set of quasi-conformal homeomorphism on $X$. The second result is that given any surface $S$ with countably many ends then there exists a hyperbolic structure $X$ such that $\text{MCG} (S)=\overline{\text{Mod}(X)}$.

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arXiv2024

Knots in circle bundles are determined by their complements

We resolve a case of the oriented knot complement conjecture by showing that knots in an orientable circle bundle $N$ over a genus $g \geq 2$ surface $S$ are determined by their complements. We apply this to the setting of canonical knots in the unit and projective tangent bundles, which are knots that are the set of tangents to a closed curve on $S$. We show that canonical knots have homeomorphic complements if and only if their shadows differ by Reidemeister moves, (de)stabilizations, loops/cusps added by transvections, and mapping classes of $S$.

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arXiv2022

Effective contraction of skinning maps

Using elementary hyperbolic geometry, we give an explicit formula for the contraction constant of the skinning map over moduli spaces of relatively acylindrical hyperbolic manifolds.

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arXiv2022

Volume bounds for the canonical lift complement of a random geodesic

Given a filling primitive geodesic curve in a closed hyperbolic surface one obtains a hyperbolic three-manifold as the complement of the curve's canonical lift to the projective tangent bundle. In this paper we give the first known lower bound for the volume of these manifolds in terms of the length of generic curves. We show that estimating the volume from below can be reduced to a counting problem in the unit tangent bundle and solve it by applying an exponential multiple mixing result for the geodesic flow.

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arXiv2022

On volumes and filling collections of multicurves

Let $S$ be a surface of negative Euler characteristic and consider a finite filling collection $Γ$ of closed curves on $S$ in minimal position. An observation of Foulon and Hasselblatt shows that $PT(S) \setminus \hatΓ$ is a finite-volume hyperbolic 3-manifold, where $PT(S)$ is the projectivized tangent bundle and $\hatΓ$ is the set of tangent lines to $Γ$. In particular, $vol(PT(S) \setminus \hatΓ)$ is a mapping class group invariant of the collection $Γ$. When $Γ$ is a filling pair of simple closed curves, we show that this volume is coarsely comparable to Weil-Petersson distance between strata in Teichmüller space. Our main tool is the study of stratified hyperbolic links $\barΓ$ in a Seifert-fibered space $N$ over $S$. For such links, the volume of $N\setminus\barΓ$ is coarsely comparable to expressions involving distances in the pants graph.

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arXiv2021

Hyperbolic limits of Cantor set complements in the sphere

Let $M$ be a hyperbolic 3-manifold with no rank two cusps admitting an embedding in $\mathbb S^3$. Then, if $M$ admits an exhaustion by $π_1$-injective sub-manifolds there exists cantor sets $C_n\subset \mathbb S^3$ such that $N_n=\mathbb S^3\setminus C_n$ is hyperbolic and $N_n\rightarrow M$ geometrically.

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arXiv2019

Hyperbolicity of links complements in Seifert fibered spaces

Let $\barγ$ be a link in a Seifert fibered space $M$ over a hyperbolic $2$-orbifolds $\mathcal O$ that projects injectively to a filling multicurve of closed geodesics $γ$ in $\mathcal O.$ We prove that the complement $M_{\barγ}$ of $\barγ$ in $M$ admits a hyperbolic structure of finite volume and give combinatorial bounds of its volume.

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arXiv2019

Hyperbolization of infinite-type 3-manifolds

We study the class $\mathcal M^B$ of 3-manifolds $M$ that have a compact exhaustion $M=\cup_{i\in\mathbb N} M_i$ satisfying: each $M_i$ is hyperbolizable with incompressible boundary and each component of $\partial M_i$ has genus at most $g= g(M)$. For manifolds in $\mathcal M^{B}$ we give necessary and sufficient topological conditions that guarantee the existence of a complete hyperbolic metric.

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arXiv2017

A locally hyperbolic 3-manifold that is not hyperbolic

We construct a locally hyperbolic 3-manifold $M_\infty$ such that $π_ 1(M_\infty)$ has no divisible subgroup. We then show that $M_\infty$ is not homeomorphic to any complete hyperbolic manifold. This answers a question of Agol [DHM06,Mar07].

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