Projective Filling Geodesic Currents

• Projective filling geodesic currents are measures on geodesics in closed surfaces that intersect every essential curve, providing a foundation for advanced geometric analysis.
• They extend Thurston’s metrics through both symmetrized and asymmetric entropy-type approaches, offering a complete framework for dynamical and rigidity phenomena.
• Length-minimizing projections associated with these currents uniquely link hyperbolic metrics to geodesic distributions, underpinning effective counting and orbit distribution results.

A projective filling geodesic current is a structure encoding the distribution of geodesics on a closed surface of negative Euler characteristic, up to positive scaling, that transversely intersects every essential closed geodesic. The space of projective filling currents, denoted $\mathbb{P}\mathcal{C}_{\mathrm{fill}}(S)$, plays a central role in higher Teichmüller theory, dynamical systems, and the geometry of moduli spaces. This locus contains and compactifies Teichmüller space under the Liouville current embedding, carries natural extensions of the Thurston metrics, and admits canonical dynamical and rigidity structures far beyond geometric surface theory.

1. Geodesic Currents and Projectivization

A geodesic current on a closed oriented surface $S$ of genus $g\ge 2$ is a $\pi_1(S)$-invariant Radon measure on the space $G$ of unoriented geodesics in the universal cover $\mathbb{H}^2$. The space $\mathcal{C}(S)$ of all geodesic currents is a metrizable, topological vector space under the weak-$*$ topology and has a natural scaling action by $\mathbb{R}_{>0}$. A closed geodesic or a measured lamination induces a discrete or continuous geodesic current, and every hyperbolic metric $X$ yields its associated Liouville current $L_X$.

The projectivization $$\mathbb{P}\mathcal{C}(S) = (\mathcal{C}(S) \setminus \{0\})/\mathbb{R}_{>0}$$ is compact, providing a natural analog to Thurston’s compactification of Teichmüller space by projective measured laminations.

A current $\mu$ is filling if its support meets every geodesic, equivalently if $i(\mu,\nu) > 0$ for all nonzero $\nu$ in $\mathcal{C}(S)$, where $i(\cdot,\cdot)$ is Bonahon's continuous, symmetric, bilinear intersection form extending the geometric intersection of closed curves. The projectivized subspace $\mathbb{P}\mathcal{C}_{\mathrm{fill}}(S)$ of filling currents is open and dense in $\mathbb{P}\mathcal{C}(S)$ and is the maximal domain for which the natural extensions of hyperbolic geometry remain meaningful (Sapir, 2022, Jyothis et al., 5 Jan 2026).

2. Extended Thurston Metrics on Projective Filling Currents

Teichmüller space $\mathcal{T}(S)$ embeds isometrically into $\mathbb{P}\mathcal{C}_{\mathrm{fill}}(S)$ via $X \mapsto [L_X]$, with $i(L_X,\alpha) = \ell_X(\alpha)$ the geodesic length function. The classical Thurston asymmetric metric on $\mathcal{T}(S)$,

$$d_{\mathrm{Th}}(X,Y) = \log \sup_{\alpha} \frac{\ell_Y(\alpha)}{\ell_X(\alpha)},$$

admits two distinct extensions to the space of projective filling currents:

• Symmetrized Thurston metric:

$$d_{\mathrm{Th}}([\mu],[\nu]) = \sup_\lambda \log \frac{i(\nu,\lambda)}{i(\mu,\lambda)} + \sup_\lambda \log \frac{i(\mu,\lambda)}{i(\nu,\lambda)},$$

which is a complete, proper metric on $\mathbb{P}\mathcal{C}_{\mathrm{fill}}(S)$ and restricts to the symmetrization of Thurston's metric on $\mathcal{T}(S)$ (Sapir, 2022).

• Asymmetric entropy-type metric:

$$d([\mu],[\nu]) = \log \left( \sup_c \frac{i(\nu,c)}{i(\mu,c)} \cdot \frac{h(\nu)}{h(\mu)} \right),$$

where $h(\mu)$ is the critical exponent, i.e., exponential growth rate of the counting function for $i(\mu,c)\le R$ over closed curves $c$. This $d$ is an asymmetric metric and extends Thurston's asymmetric metric, satisfying $d([L_X],[L_Y]) = d_{\mathrm{Th}}(X,Y)$ (Jyothis et al., 5 Jan 2026, Sapir, 2022).

No quasi-isometric retraction from $\mathbb{P}\mathcal{C}_{\mathrm{fill}}(S)$ to $\mathcal{T}(S)$ exists for either metric (Sapir, 2022, Jyothis et al., 5 Jan 2026), in sharp contrast to the boundary behavior of the classical setting.

3. Length-Minimizing Projections and Rigidity

Given a projective filling current $[\mu]$, the length-minimizing projection

$$\pi : \mathbb{P}\mathcal{C}_{\mathrm{fill}}(S) \to \mathcal{T}(S)$$

is defined by

$$\pi([\mu]) = \operatorname{argmin}_{X \in \mathcal{T}(S)} i(\mu,X).$$

Here $i(\mu,X) := \ell_X(\mu)$ is the total $\mu$-length in the metric $X$. This projection is continuous, proper, and equivariant with respect to the mapping class group (Hensel et al., 2021, Sapir, 2022, Sapir, 2022). The minimizer is unique due to strict convexity of $X \mapsto i(\mu,X)$ along Weil–Petersson geodesics.

Key properties include:

• Fibers over points in $\mathcal{T}(S)$ are compact, but their diameters in the Thurston metric can diverge in the thin parts.
• For measured laminations $\lambda_1+\lambda_2$ with filling support, the projection corresponds to the metric minimizing $\ell(\lambda_1)+\ell(\lambda_2)$ (Kerckhoff's line-of-minima).
• As a filling current degenerates towards a uniquely ergodic lamination, its image under $\pi$ converges to the corresponding projective lamination in Thurston’s boundary.

Moreover, recent work shows that the horofunction compactification of $(\mathbb{P}\mathcal{C}_{\mathrm{fill}}(S), d)$ can be identified with the full space $\mathbb{P}\mathcal{C}(S)$, and isometric rigidity holds: surfaces of different genera do not yield isometric spaces $(\mathbb{P}\mathcal{C}_{\mathrm{fill}}(S), d)$ (Jyothis et al., 5 Jan 2026).

4. Dynamical and Asymptotic Aspects

The metric structures associated with projective filling currents yield deep counting results and entropy invariants. For a filling current $\mu$, the critical exponent $\delta(\mu)$ is defined as

$$\delta(\mu) = \limsup_{R\to\infty} \frac{1}{R} \log \#\{ [c] : i(\mu, c) \le R \},$$

which coincides with the exponential growth of intersections. For the Liouville current $L_X$ of a hyperbolic surface $X$, one has $\delta(L_X) = 1$ (Glorieux, 2017).

Scaling behaves as $\delta(t\mu) = \delta(\mu)/t$, so the critical exponent descends to a proper, continuous function on projective filling currents, serving as a natural entropy functional throughout $\mathbb{P}\mathcal{C}_{\mathrm{fill}}(S)$. This functional generalizes the volume entropy of a hyperbolic metric to the infinite-dimensional moduli space of projective filling currents, with minimizing points corresponding to hyperbolic metrics.

Furthermore, the metric $d_\mu$ on the universal cover defined by

$$d_\mu(x, y) = \mu(\text{geodesics transversely intersecting } [x, y])$$

is a proper, $\pi_1(S)$-invariant Gromov-hyperbolic distance if and only if $\mu$ is filling, leading to highly nontrivial geometric and dynamical structures (Glorieux, 2017).

5. Counting, Orbit Distribution, and Measure Rigidity

Projective filling geodesic currents govern the asymptotic behavior of moduli space and mapping class group dynamics:

• For any filling compactly supported current $\alpha$ and any positive, homogeneous, continuous functional $f$, the number of mapping classes $\phi$ with $f(\phi(\alpha)) \leq L$ grows as $L^d$ as $L \to \infty$, with $d = \dim \mathbb{P}\mathcal{C}(S) = 6g-6+2n$, and explicit constants in terms of Thurston measure on measured laminations (Rafi et al., 2017, Arana-Herrera, 2021).
• For counting filling closed curve orbits of a given topological type under the mapping class group with respect to intersection with a filling current, a power-saving error term is achieved and the empirical measures on orbit representatives equidistribute in $\mathbb{P}\mathcal{C}(S)$ for large $L$ (Arana-Herrera, 2021).

The input of filling and projectivization is crucial: sublevel sets $\{f \le 1\}$ are compact in $\mathbb{P}\mathcal{C}(S)$ due to homogeneity, and the filling property ensures positivity under intersection, disallowing degenerate or tangential behaviors.

6. Extensions and Broader Frameworks

The theory of projective filling geodesic currents admits broad generalizations:

• The asymmetric metric $d$ on $\mathbb{P}\mathcal{C}_{\mathrm{fill}}(S)$ extends to the space of metric structures on any nonelementary Gromov hyperbolic group, and further to hyperbolic potentials, capturing and generalizing the metric geometry of Hitchin representations, Anosov representations, and length-spectrum metrics (Jyothis et al., 5 Jan 2026).
• This universality connects the geometry of currents on surfaces, higher representation spaces, and group-theoretic metric structures, identifying projective filling currents as the natural geometric boundary in a wide array of moduli problems.

7. Summary Table: Key Structures for Projective Filling Geodesic Currents

Concept	Notation/Symbol	Defining Property
Space of geodesic currents	$\mathcal{C}(S)$	$\pi_1(S)$-invariant Radon measures on unoriented geodesics
Projectivized filling currents	$\mathbb{P}\mathcal{C}_{\mathrm{fill}}(S)$	Projectivization of currents intersecting every essential geodesic
Intersection pairing	$i(\mu,\nu)$	Continuous symmetric bilinear extension of geometric intersection
Symmetrized Thurston metric	$d_{Th}([\mu],[\nu])$	$$\sup_\lambda \log\frac{i(\nu,\lambda)}{i(\mu,\lambda)}+\sup_\lambda\log\frac{i(\mu,\lambda)}{i(\nu,\lambda)}$$
Length-minimizing projection	$\pi([\mu])$	Unique $X\in\mathcal{T}(S)$ minimizing $i(\mu,X)$
Critical exponent	$\delta(\mu)$	$$\limsup_{R\to\infty} \frac{1}{R}\log\#\{[c]:i(\mu,c)\le R\}$$

These structures provide a dense algebraic, metric, and dynamical framework for the geometry of surfaces, orbit statistics, and moduli problems beyond the setting of Teichmüller and measured lamination spaces.

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References:

(Glorieux, 2017, Rafi et al., 2017, Arana-Herrera, 2021, Hensel et al., 2021, Sapir, 2022, Sapir, 2022, Jyothis et al., 5 Jan 2026)