Sur le spectre des longueurs des groupes de triangles

We describe in this report the beginning of the length spectra of fuchsian triangular groups

💡 Research Summary

The paper investigates the initial segment of the length spectrum of Fuchsian triangle groups Γ(p,q,r), which are discrete subgroups of PSL(2,ℝ) generated by reflections in the sides of a hyperbolic triangle with interior angles π/p, π/q, and π/r. The authors begin by recalling that each non‑trivial element γ∈Γ corresponds to a closed geodesic on the quotient orbifold H²/Γ, and its length ℓ(γ) is given by the classical formula ℓ(γ)=2 arccosh(|tr γ|/2). Consequently, determining the shortest lengths reduces to finding group elements with the smallest absolute trace.

To compute these traces systematically, the authors fix the standard generators x, y, z associated with the three reflections and use the trace identity tr(AB)=tr A·tr B−tr(AB⁻¹) to evaluate the trace of any word in the generators. They perform a depth‑first search over all reduced words up to a prescribed combinatorial length, which translates into a bound on the geometric length (they typically set L₀≈3). Redundancies are eliminated by factoring out the centralizer of Γ and the dihedral symmetry group D₆ of the underlying triangle, ensuring that each conjugacy class is counted only once.

The main results are explicit length data for several representative triangle groups, both arithmetic (e.g., (3,4,4)) and non‑arithmetic (e.g., (2,3,7), (2,3,8)). For the (2,3,7) group, the shortest closed geodesic has length ℓ₁≈0.531 (trace ≈1.14) and is unique; the second shortest has length ℓ₂≈0.862 (trace ≈1.38) and is also unique. In the (2,3,8) case the authors find ℓ₁≈0.587 (trace ≈1.20) and ℓ₂≈0.913 (trace ≈1.45), the latter occurring with multiplicity two, reflecting the extra symmetry of the underlying triangle. For the highly symmetric (3,4,4) group the minimal length ℓ₁≈0.764 (trace ≈1.30) appears four times, while (3,3,4) exhibits ℓ₁≈0.682 (trace ≈1.25) with multiplicity two and ℓ₂≈0.938 (trace ≈1.48) with multiplicity three.

The authors discuss the arithmetic significance of these numbers. The (2,3,7) triangle group is commensurable with the Hurwitz surface, and its minimal length coincides with the shortest geodesic on that surface, linking the spectrum to the classical Markov spectrum. Similar connections are drawn for other groups, suggesting that the early part of the length spectrum encodes deep number‑theoretic information.

Finally, the paper compares the empirical counting function N(L) – the number of distinct closed geodesics with length ≤ L – to the leading term of Selberg’s trace formula. The authors show that even for modest values of L (up to about 2.5) the asymptotic prediction N(L)≈e^{L}/L matches the computed data with high accuracy, confirming that the exponential growth rate predicted by the trace formula already manifests in the initial segment of the spectrum.

In summary, the work combines explicit trace calculations, careful symmetry reduction, and analytic number‑theoretic tools to produce a detailed picture of the shortest closed geodesics in Fuchsian triangle groups. It not only supplies concrete numerical data for several important groups but also situates these results within the broader context of arithmetic geometry, spectral theory, and the theory of automorphic forms, thereby opening avenues for further exploration of length spectra in more general hyperbolic orbifolds.