arXiv subjects
Publications and source records attributed to David Fisac.
On the once-punctured torus, we give an exact formula for the number of curves in any given mapping class group orbit of given word length. This settles a conjecture of Chas in [Cha16].
We study infinite-type hyperbolic surfaces with discrete length spectrum. In this setup, we show that Sunada's method can only produce finite isospectral families, but that there is no bound on the cardinality of an isospectral family (for any infinite-type surface without planar ends). We then prove that, given any infinite-genus surface satisfying an additional topological assumption, any finite group can be realized as isometry group of a hyperbolic structure with discrete length spectrum.
We use the geometric reformulation of Markov's uniqueness conjecture in terms of the simple length spectrum on the modular torus to rewrite the conjecture in combinatorial terms by explicitly describing this set of lengths.
We classify closed curves on a once-punctured torus with a single selfintersection from a combinatorial perspective. We then use the classification to determine the number of closed curves with given word-length and with zero, one, and arbitrary self-intersections.
We explore for compact Riemannian surfaces whose boundary consists of a single closed geodesic the relationship between orthospectrum and boundary length. More precisely, we establish a uniform lower bound on the boundary length in terms of the orthospectrum when fixing a metric invariant of the surface related to the classical notion of volume entropy. This inequality can be thought of as a Riemannian analog of Basmajian's identity for hyperbolic surfaces.