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Federica Fanoni

Publications and source records attributed to Federica Fanoni.

16 recordsLinked to original sources
arXiv2026

Isospectrality and isometry groups for infinite-type hyperbolic surfaces with discrete length spectrum

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.

math.GT↗
arXiv2026

Approximating stable translation lengths on fine curve graphs

We study the stable translation length of homeomorphisms of a surface acting on the fine nonseparating curve graph and compare it to the stable translation lengths of its finite approximations - mapping classes relative to a finite invariant set - acting on the nonseparating curve graph. We prove that the stable translation length of a homeomorphism with a dense set of periodic points is the supremum of the stable translation lengths of its approximations, and that the stable translation length is preserved under cell-like extensions. We deduce that homotopically triv ial homeomorphisms of the torus have stable translation length which is the supremum of the stable translation lengths of their finite approximations. We show that the supremum is not always a maximum, by proving that the stable translation length of a mapping class acting on the nonseparating curve graph is rational.

math.DS↗
arXiv2024

Orthosystoles and orthokissing numbers

For hyperbolic surfaces with geodesic boundary, we study the orthosystole, i.e. the length of a shortest essential arc from the boundary to the boundary. We recover and extend work by Bavard completely characterizing the surfaces maximizing the orthosystole in the case of a single boundary component. For multiple boundary components, we construct surfaces with large orthosystole and show that their orthosystole grows, as the genus goes to infinity, at the same rate as Bavard's upper bound.

math.GT↗
arXiv2023

Towards Nielsen-Thurston classification for surfaces of infinite type: well-tempered homeomorphisms

We introduce and study tempered mapping classes of surfaces of infinite type. These are maps for which curves under iteration do not accumulate onto geodesic laminations with non-proper leaves, but only on unions of possibly intersecting curves or proper lines. Assuming an additional finiteness condition on the accumulation set, we prove a Nielsen-Thurston-type classification theorem. We prove that for such maps there is a canonical decomposition of the surface into invariant subsurfaces on which the first return is either periodic or a translation.

math.GT↗
arXiv2023

Multitwists in big mapping class groups

We show that the closure of the compactly supported mapping class group of an infinite-type surface is not generated by the collection of multitwists (i.e. products of powers of twists about disjoint non-accumulating curves).

math.GT↗
arXiv2021

Homeomorphic subsurfaces and the omnipresent arcs

In this article, we are concerned with various aspects of arcs on surfaces. In the first part, we deal with topological aspects of arcs and their complements. We use this understanding, in the second part, to construct interesting actions of the mapping class group on a subgraph of the arc graph. This subgraph naturally emerges from a new characterisation of infinite-type surfaces in terms of homeomorphic subsurfaces.

math.GT↗
arXiv2020

Isospectral hyperbolic surfaces of infinite genus

We show that any infinite-type surface without planar ends admits arbitrarily large families of length isospectral hyperbolic structures. If the surface has infinite genus and its space of ends is self-similar, we construct an uncountable family of isospectral and quasiconformally distinct hyperbolic structures.

math.GT↗
arXiv2019

Big mapping class groups acting on homology

We study the action of (big) mapping class groups on the first homology of the corresponding surface. We give a precise characterization of the image of the induced homology representation.

math.GT↗
arXiv2017

Basmajian-type inequalities for maximal representations

For suitable metrics on the locally symmetric space associated to a maximal representation, we prove inequalities between the length of the boundary and the lengths of orthogeodesics that generalize the classical Basmajian's identity from Teichmueller theory. Any equality characterizes diagonal embeddings.

math.GT↗
arXiv2016

Graphs of curves on infinite-type surfaces with mapping class group actions

We study when the mapping class group of an infinite-type surface $S$ admits an action with unbounded orbits on a connected graph whose vertices are simple closed curves on $S$. We introduce a topological invariant for infinite-type surfaces that determines in many cases whether there is such an action. This allows us to conclude that, as non-locally compact topological groups, many big mapping class groups have nontrivial coarse geometry in the sense of Rosendal.

math.GT↗
arXiv2016

Mapping class group orbits of curves with self-intersections

We study mapping class group orbits of homotopy and isotopy classes of curves with self-intersections. We exhibit the asymptotics of the number of such orbits of curves with a bounded number of self-intersections, as the complexity of the surface tends to infinity. We also consider the minimal genus of a subsurface that contains the curve. We determine the asymptotic number of orbits of curves with a fixed minimal genus and a bounded self-intersection number, as the complexity of the surface tends to infinity. As a corollary of our methods, we obtain that most curves that are homotopic are also isotopic. Furthermore, using a theorem by Basmajian, we get a bound on the number of mapping class group orbits on a given a hyperbolic surface that can contain short curves. For a fixed length, this bound is polynomial in the signature of the surface. The arguments we use are based on counting embeddings of ribbon graphs.

math.GT↗
arXiv2016

Simplicial embeddings between multicurve graphs

We study some graphs associated to a surface, called k-multicurve graphs, which interpolate between the curve complex and the pants graph. Our main result is that, under certain conditions, simplicial embeddings between multicurve graphs are induced by $π_1$-injective embeddings of the corresponding surfaces. We also prove the rigidity of the multicurve graphs.

math.GT↗
arXiv2015

Systoles and kissing numbers of finite area hyperbolic surfaces

We study the number and the length of systoles on complete finite area orientable hyperbolic surfaces. In particular, we prove upper bounds on the number of systoles that a surface can have (the so-called kissing number for hyperbolic surfaces). Our main result is a bound which only depends on the topology of the surface and which grows subquadratically in the genus.

math.GT↗
arXiv2015

Filling sets of curves on punctured surfaces

We study filling sets of simple closed curves on punctured surfaces. In particular we study lower bounds on the cardinality of sets of curves that fill and that pairwise intersect at most k times on surfaces with given genus and number of punctures. We are able to establish orders of growth for even k and show that for odd k the orders of growth behave differently. We also study the corresponding questions when one requires that the curves be represented as systoles on hyperbolic complete finite area surfaces.

math.GT↗
arXiv2014

The maximum injectivity radius of hyperbolic orbifolds

For two-dimensional orientable hyperbolic orbifolds, we show that the radius of a maximal embedded disk is greater or equal to an explicit constant ρ_T, with equality if and only if the orbifold is a sphere with three cone points of order 2, 3 and 7.

math.GT↗