arXiv subjects

Sara Maloni

Publications and source records attributed to Sara Maloni.

17 recordsLinked to original sources

arXiv2026

Dynamics and geometry of character varieties for surface groups

The problem of classifying geometric structures on manifolds is very much related to the discussion of the automorphism groups actions on character varieties, which are spaces of equivalence classes of representations. In this chapter we survey some results on this topic, mostly focusing on representations of surface groups (both in the orientable and non-orientable cases) and free groups. An important principle in the study of the dynamics on character varieties $X=X(\pi_1(S),G)$ for surface groups $\pi_1(S)$ is the following dichotomy: when the target group $G$ is compact, $X$ has nontrivial homotopy type, and the action of the mapping class group is chaotic; whereas when the target group $G$ is non-compact, $X$ contains contractible sets on which the mapping class group acts properly. We will expand on this dichotomy in various cases. After introducing the necessary background, we will discuss representations into $\mathsf{PSL}_2(\mathbb{R})$ and $\mathsf{PGL}_2(\mathbb{R})$, discussing the number of connected components, the geometric properties (Bowditch question), the dynamics (Goldman conjecture) and some components with an `exotic' behaviour (Deroin-Tholozan representations). We will also underline how the theory for representations of fundamental groups of orientable closed hyperbolizable surfaces needs to be adapted when one considers surfaces with punctures or non-orientable surfaces. We will then discuss representations into compact groups, where we will discuss mostly ergodicity results in various settings, and some non-ergodicity results at the end. Thirdly, we will consider representations in $\mathsf{PSL}_2(\mathbb{C})$. We will discuss convex-cocompact representations, primitive-stable and Bowditch representations and their relationship. Finally, we will describe how some of the results mentioned can be generalized for representations into higher-rank Lie groups.

math.GT↗

arXiv2025

Topology of the space of $d$-pleated surfaces

Given a maximal geodesic lamination $\lambda$ on a closed oriented surface $S$ of genus $g$, the space of $d$-pleated surfaces with pleating locus $\lambda$ is an open subset of $\mathrm{Hom}(\pi_1(S),\mathsf{PGL}_d(\mathbb{C}))$ obtained by applying generalized bending along $\lambda$ to Hitchin representations. When $d=2$, one recovers abstract pleated surfaces in $\mathbb{H}^3$. In this paper, we study the topology of the space $\mathfrak{R}(\lambda,d)$ of conjugacy classes of $d$-pleated surfaces with pleating locus $\lambda$. Firstly, we prove that $\mathfrak{R}(\lambda,d)$ is real-analytically diffeomorphic to $\mathbb{R}^{(d^2-1)(2g-2)}\times(\mathbb{R}/2\pi\mathbb{Z})^{(d^2-1)(2g-2)}\times \mathbb{Z}_d$, where $\mathbb{Z}_d$ denotes the finite cyclic group of order $d$. Furthermore, we show that each connected component of the space of conjugacy classes in $\mathrm{Hom}(\pi_1(S),\mathsf{PGL}_d(\mathbb{C}))$ contains exactly one component of $\mathfrak{R}(\lambda,d)$.

math.GT↗

arXiv2025

Constructing reducibly geometrically finite subgroups of the mapping class group

In this article, we consider qualified notions of geometric finiteness in mapping class groups called parabolically geometrically finite (PGF) and reducibly geometrically finite (RGF). We examine several constructions of subgroups and determine when they produce a PGF or RGF subgroup. These results provide a variety of new examples of PGF and RGF subgroups. Firstly, we consider the right-angled Artin subgroups constructed by Koberda and Clay--Leininger--Mangahas, which are generated by high powers of given elements of the mapping class group. We give conditions on the supports of these elements that imply the resulting right-angled Artin subgroup is RGF. Secondly, we prove combination theorems which provide conditions for when a collection of reducible subgroups, or sufficiently deep finite-index subgroups thereof, generate an RGF subgroup.

math.GT↗

arXiv2024

Dynamics on the SU(2,1)-character variety of the one-holed torus

We study the relative SU(2,1)-character varieties of the one-holed torus, and the action of the mapping class group on them. We use an explicit description of the character variety of the free group of rank two in SU(2,1) in terms of traces, which allow us to describe the topology of the character variety. We then combine this description with a generalization of the Farey graph adapted to this new combinatorial setting, using ideas introduced by Bowditch. Using these tools, we can describe an open domain of discontinuity for the action of the mapping class group which strictly contains the set of convex cocompact characters, and we give several characterizations of representations in this domain.

math.GT↗

arXiv2023

$d$-pleated surfaces and their shear-bend coordinates

In this article, we single out representations of surface groups into $\mathsf{PSL}_d(\mathbb{C})$ which generalize the well-studied family of pleated surfaces into $\mathsf{PSL}_2(\mathbb{C})$. Our representations arise as sufficiently generic $\lambda$-Borel Anosov representations, which are representations that are Borel Anosov with respect to a maximal geodesic lamination $\lambda$. For fixed $\lambda$ and $d$, we provide a holomorphic parametrization of the space $\mathcal{R}(\lambda,d)$ of $(\lambda,d)$-pleated surfaces which extends both work of Bonahon for pleated surfaces and Bonahon and Dreyer for Hitchin representations.

math.GT↗

arXiv2023

Fiber bundles associated with Anosov representations

Anosov representations $ρ$ of a hyperbolic group $Γ$ into a semisimple Lie group $G$ are known to admit cocompact domains of discontinuity in flag varieties $G/Q$, endowing the compact quotient manifolds $M_ρ$ with a $(G,G/Q)$-structure. In general the topology of $M_ρ$ can be quite complicated. In this article, we consider the case when $Γ$ is the fundamental group of a closed (real or complex) hyperbolic manifold $N$ and $ρ$ is a deformation of a (twisted) lattice embedding $Γ\to \mathrm{Isom}(\mathbb H_\mathbb K) \to G$ through Anosov representations. We prove that, in this situation, $M_ρ$ is alway a smooth fiber bundle over $N$. Determining the topology of the fiber seems hard in general. The second part of the paper focuses on the special case when $N$ is a surface, $ρ$ a quasi-Hitchin representation into $\mathrm{Sp}(4,\mathbb C)$, and $M_ρ$ is modelled on the space of complex Lagrangians in $\mathbb C^4$. We show that, in this case, the fiber is homeomorphic to $\mathbb{CP}^2 \sharp \overline{\mathbb{CP}^2}$.

math.GT↗

arXiv2020

Geometric limits of cyclic subgroups of SO_0(1, k+1) and SU(1, k+1)

We study geometric limits of convex-cocompact cyclic subgroups of the rank 1 groups SO_0(1, k+1) and SU(1, k+1). We construct examples of sequences of subgroups of such groups G that converge algebraically and whose geometric limit strictly contains the algebraic limit, thus generalizing the example first described by Jorgensen for subgroups of SO_0(1,3). We also give necessary and sufficient conditions for a subgroup of SO_0(1, k+1) to arise as geometric limit of a sequence of cyclic subgroups. We then discuss generalizations of such examples to sequence of representations of free groups, and applications of our constructions in that setting.

math.GT↗

arXiv2019

Quasicircles and width of Jordan curves in $\mathbb{CP}^1$

We study a notion of "width" for Jordan curves in $\mathbb{CP}^1$, paying special attention to the class of quasicircles. The width of a Jordan curve is defined in terms of the geometry of its convex hull in hyperbolic three-space. A similar invariant in the setting of anti de Sitter geometry was used by Bonsante-Schlenker to characterize quasicircles amongst a larger class of Jordan curves in the boundary of anti de Sitter space. By contrast to the AdS setting, we show that there are Jordan curves of bounded width which fail to be quasicircles. However, we show that Jordan curves with small width are quasicircles.

math.GT↗

arXiv2019

The induced metric on the boundary of the convex hull of a quasicircle in hyperbolic and anti de Sitter geometry

Celebrated work of Alexandrov and Pogorelov determines exactly which metrics on the sphere are induced on the boundary of a compact convex subset of hyperbolic three-space. As a step toward a generalization for unbounded convex subsets, we consider convex regions of hyperbolic three-space bounded by two properly embedded disks which meet at infinity along a Jordan curve in the ideal boundary. In this setting, it is natural to augment the notion of induced metric on the boundary of the convex set to include a gluing map at infinity which records how the asymptotic geometry of the two surfaces compares near points of the limiting Jordan curve. Restricting further to the case in which the induced metrics on the two bounding surfaces have constant curvature $K \in [-1,0)$ and the Jordan curve at infinity is a quasicircle, the gluing map is naturally a quasisymmetric homeomorphism of the circle. The main result is that for each value of $K$, every quasisymmetric map is achieved as the gluing map at infinity along some quasicircle. We also prove analogous results in the setting of three-dimensional anti de Sitter geometry. Our results may be viewed as universal versions of the conjectures of Thurston and Mess about prescribing the induced metric on the boundary of the convex core of quasifuchsian hyperbolic manifolds and globally hyperbolic anti de Sitter spacetimes.

math.GT↗

arXiv2019

C-gluing construction and slices of quasi-Fuchsian space

Given a pants decomposition $\mathcal{PC} = \{γ_1, \ldots, γ_ξ\}$ on a hyperbolizable surface $Σ$ and a vector $\underline{c} = (c_1, \ldots, c_ξ) \in \mathbb{R}_+^ξ$, we describe a plumbing construction which endows $Σ$ with a complex projective structure for which the associated holonomy representation $ρ$ is quasi-Fuchsian and for which $\ell_ρ(γ_i) = c_i$. When $\underline{c} \to \underline{0} = (0, \ldots, 0)$ this construction limits to Kra's plumbing construction. In addition, when $Σ= Σ_{1,1}$, the holonomy representations of these structures belong to the `linear slice' of quasi-Fuchsian space $\mathrm{QF}(Σ)$ defined by Komori and Parkonnen. We discuss some conjectures for these slices suggested by the pictures we created in joint work with Yamashita.

math.GT↗

arXiv2018

On type-preserving representations of the thrice punctured projective plane group

In this paper we consider type-preserving representations of the fundamental group of the three--holed projective plane into $\mathrm{PGL}(2, \R) =\mathrm{Isom}(\HH^2)$ and study the connected components with non-maximal euler class. We show that in euler class zero for all such representations there is a one simple closed curve which is non-hyperbolic, while in euler class $\pm 1$ we show that there are $6$ components where all the simple closed curves are sent to hyperbolic elements and $2$ components where there are simple closed curves sent to non-hyperbolic elements. This answer a question asked by Brian Bowditch. In addition, we show also that in most of these components the action of the mapping class group on these non-maximal component is ergodic. In this work, we use an extension of Kashaev's theory of decorated character varieties to the context of non-orientable surfaces.

math.GT↗

arXiv2016

Higher signature Delaunay decompositions

A Delaunay decomposition is a cell decomposition in R^d for which each cell is inscribed in a Euclidean ball which is empty of all other vertices. This article introduces a generalization of the Delaunay decomposition in which the Euclidean balls in the empty ball condition are replaced by other families of regions bounded by certain quadratic hypersurfaces. This generalized notion is adaptable to geometric contexts in which the natural space from which the point set is sampled is not Euclidean, but rather some other flat semi-Riemannian geometry, possibly with degenerate directions. We prove the existence and uniqueness of the decomposition and discuss some of its basic properties. In the case of dimension d = 2, we study the extent to which some of the well-known optimality properties of the Euclidean Delaunay triangulation generalize to the higher signature setting. In particular, we describe a higher signature generalization of a well-known description of Delaunay decompositions in terms of the intersection angles between the circumscribed circles.

cs.CG↗

arXiv2015

On the character variety of the three-holed projective plane

We study the (relative) SL(2,C) character varieties of the three-holed projective plane and the action of the mapping class group on them. We describe a domain of discontinuity for this action, which strictly contains the set of primitive stable representations defined by Minsky, and also the set of convex-cocompact characters. We consider the relationship with the previous work of the authors and S. P. Tan on the character variety of the four-holed sphere.

math.GT↗

arXiv2014

Polyhedra inscribed in a quadric

We study convex polyhedra in three-space that are inscribed in a quadric surface. Up to projective transformations, there are three such surfaces: the sphere, the hyperboloid, and the cylinder. Our main result is that a planar graph $Γ$ is realized as the $1$-skeleton of a polyhedron inscribed in the hyperboloid or cylinder if and only if $Γ$ is realized as the $1$-skeleton of a polyhedron inscribed in the sphere and $Γ$ admits a Hamiltonian cycle. Rivin characterized convex polyhedra inscribed in the sphere by studying the geometry of ideal polyhedra in hyperbolic space. We study the case of the hyperboloid and the cylinder by parameterizing the space of convex ideal polyhedra in anti-de Sitter geometry and in half-pipe geometry. Just as the cylinder can be seen as a degeneration of the sphere and the hyperboloid, half-pipe geometry is naturally a limit of both hyperbolic and anti-de Sitter geometry. We promote a unified point of view to the study of the three cases throughout.

math.DG↗

arXiv2013

On the character variety of the four-holed sphere

We study the (relative) SL(2,C) character varieties of the four-holed sphere and the action of the mapping class group on it. We describe a domain of discontinuity for this action, and, in the case of real characters, show that this domain of discontinuity may be non-empty on the components where the relative euler class is non-maximal.

math.GT↗

arXiv2011

The asymptotic directions of pleating rays in the Maskit embedding

This article was born as a generalisation of the analysis made by Series, where she made the first attempt to plot a deformation space of Kleinian group of more than 1 complex dimension. We use the Top Terms' Relationship proved by the author and Series to determine the asymptotic directions of pleating rays in the Maskit embedding of a hyperbolic surface S as the bending measure of the `top' surface in the boundary of the convex core tends to zero. The Maskit embedding M of a surface S is the space of geometrically finite groups on the boundary of quasifuchsian space for which the `top' end is homeomorphic to S, while the `bottom' end consists of triply punctured spheres, the remains of S when the pants curves have been pinched. Given a projective measured lamination l on S, the pleating ray P is the set of groups in M for which the bending measure of the top component of the boundary of the convex core of the associated 3-manifold is in the projective class of l.

math.GT↗

arXiv2010

Top terms of polynomial traces in Kra's plumbing construction

Let $Σ$ be a surface of negative Euler characteristic together with a pants decomposition $¶$. Kra's plumbing construction endows $Σ$ with a projective structure as follows. Replace each pair of pants by a triply punctured sphere and glue, or `plumb', adjacent pants by gluing punctured disk neighbourhoods of the punctures. The gluing across the $i^{th}$ pants curve is defined by a complex parameter $τ_i \in \C$. The associated holonomy representation $ρ: π_1(Σ) \to PSL(2,\C)$ gives a projective structure on $Σ$ which depends holomorphically on the $τ_i$. In particular, the traces of all elements $ρ(γ), γ\in π_1(Σ)$, are polynomials in the $τ_i$. Generalising results proved in previous papers for the once and twice punctured torus respectively, we prove a formula giving a simple linear relationship between the coefficients of the top terms of $ρ(γ)$, as polynomials in the $τ_i$, and the Dehn-Thurston coordinates of $γ$ relative to $¶$. This will be applied elsewhere to give a formula for the asymptotic directions of pleating rays in the Maskit embedding of $Σ$ as the bending measure tends to zero.

math.GN↗