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Joan Porti

Publications and source records attributed to Joan Porti.

At least 19 recordsLinked to original sources
arXiv2026

Knots with large character varieties

We study knots whose $\mathrm{SL}_2(\mathbb{C})$-character varieties have a component of dimension greater than one. We call such knots $\mathcal{X}$-large and introduce two diagrammatic constructions that produce $\mathcal{X}$-large knots. The first construction uses split link diagrams and rational tangle replacements, providing a topological explanation for most $\mathcal{X}$-large knots observed in knot tables. The second construction is based on braids and orientation-reversing involutions, and is motivated by a detailed analysis of the knot $10_{123}$, also known as the Turk's head knot $Th(3,5)$. In particular, this approach applies to Turk's head knots $Th(p,q)$ with $p$ and $q$ odd, leading us to conjecture that all such knots are $\mathcal{X}$-large. In doing so, we also present a non-orientable analogue of Thurston's theorem giving a lower bound on the dimension of character varieties of non-orientable 3-manifolds.

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arXiv2026

Representations of the modular group into the isometries of SL(3,R)/SO(3)

We describe the topology of the variety of characters of the modular group $\mathrm{PSL}_2(\mathbb{Z})$ into the isometry group of the symmetric space $\mathrm{SL}_3(\mathbb{R})/\mathrm{SO}(3)$. This topology is determined in terms of the fixed point sets of the generators of the modular group. We also find Anosov representations on each nontrivial component.

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arXiv2025

Projective deformations of hyperbolic 3-orbifolds with turnover ends

We study projective deformations of (topologically finite) hyperbolic 3-orbifolds whose ends have turnover cross section. These deformations are examples of projective cusp openings, meaning that hyperbolic cusps are deformed in the projective setting such that they become totally geodesic generalized cusps with diagonal holonomy. We find that this kind of structure is the only one that can arise when deforming hyperbolic turnover cusps, and that turnover funnels remain totally geodesic. Therefore, we argue that, under no infinitesimal rigidity assumptions, the deformed projective 3-orbifold remains properly convex. Additionally, we give a complete description of the character variety of the turnover $S^2(3,3,3)$ in SL(4,R).

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arXiv2025

Deforming reducible representations of surface and 2-orbifold groups

For a compact 2-orbifold with negative Euler characteristic $\mathcal O^2$, the variety of characters of $π_1(\mathcal O^2)$ in $\mathrm{SL}_{n}(\mathbb R)$ is a non-singular manifold at $\mathbb C$-irreducible representations. In this paper we prove that when a $\mathbb C$-irreducible representation of $π_1(\mathcal O^2)$ in $\mathrm{SL}_{n}(\mathbb R)$ is viewed in $\mathrm{SL}_{n+1}(\mathbb R)$, then the variety of characters is singular, and we describe the singularity.

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arXiv2024

New geometric structures on 3-manifolds: surgery and generalized geometry

Cosymplectic and normal almost contact structures are analogues of symplectic and complex structures that can be defined on 3-manifolds. Their existence imposes strong topological constraints. Generalized geometry offers a natural common generalization of these two structures: $B_3$-generalized complex structures. We prove that any closed orientable 3-manifold admits such a structure, which can be chosen to be stable, that is, generically cosymplectic up to generalized diffeomorphism.

math.DG↗
arXiv2023

The scheme of characters in SL 2

The aim of this article is to study the SL(2,C)-character scheme of a finitely generated group. Given a presentation of a finitely generated group $Γ$, we give equations defining the coordinate ring of the scheme of SL(2,C)-characters of $Γ$ (finitely many equations when $Γ$ is finitely presented). We also study the scheme of abelian and nonsimple representations and characters. Finally we apply our results to study the SL(2,C)-character scheme of the Borromean rings.

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arXiv2023

Morse actions of discrete groups on symmetric spaces: Local-to-global principle

Our main result is a local-to-global principle for Morse quasigeodesics, maps and actions. As an application of our techniques we show algorithmic recognizability of Morse actions and construct Morse ``Schottky subgroups'' of higher rank semisimple Lie groups via arguments not based on Tits' ping-pong. Our argument is purely geometric and proceeds by constructing equivariant Morse quasiisometric embeddings of trees into higher rank symmetric spaces.

math.DG↗
arXiv2022

The deformation space of non-orientable hyperbolic 3-manifolds

We consider non-orientable hyperbolic 3-manifolds of finite volume $M^3$. When $M^3$ has an ideal triangulation $Δ$, we compute the deformation space of the pair $(M^3, Δ)$ (its Neumann Zagier parameter space). We also determine the variety of representations of $π_1(M^3)$ in $\mathrm{Isom}(\mathbb{H}^3)$ in a neighborhood of the holonomy. As a consequence, when some ends are non-orientable, there are deformations from the variety of representations that cannot be realized as deformations of the pair $(M^3, Δ)$. We also discuss the metric completion of these structures and we illustrate the results on the Gieseking manifold.

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arXiv2021

The adjoint Reidemeister torsion for the connected sum of knots

Let $K$ be the connected sum of knots $K_1,\ldots,K_n$. It is known that the $\mathrm{SL}_2(\mathbb{C})$-character variety of the knot exterior of $K$ has a component of dimension $\geq 2$ as the connected sum admits a so-called bending. We show that there is a natural way to define the adjoint Reidemeister torsion for such a high-dimensional component and prove that it is locally constant on a subset of the character variety where the trace of a meridian is constant. We also prove that the adjoint Reidemeister torsion of $K$ satisfies the vanishing identity if each $K_i$ does so.

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arXiv2021

Projective structures on a hyperbolic 3-orbifold

We compute and analyse the moduli space of those real projective structures on a hyperbolic 3-orbifold that are modelled on a single ideal tetrahedron in projective space. Parameterisations are given in terms of classical invariants, traces, and geometric invariants, cross ratios.

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arXiv2020

Holomorphic volume forms on representation varieties of surfaces with boundary

For closed and oriented hyperbolic surfaces, a formula of Witten establishes an equality between two volume forms on the space of representations of the surface in a semisimple Lie group. One of the forms is a Reidemeister torsion, the other one is the power of the Atiyah-Bott-Goldman symplectic form. We introduce an holomorphic volume form on the space of representations of the circle, so that, for surfaces with boundary, it appears as peripheral term in the generalization of Witten's formula. We compute explicit volume and symplectic forms for some simple surfaces and for the Lie group SL(N,C).

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arXiv2020

Dimension of representation and character varieties for two and three-orbifolds

We consider varieties of representations and characters of 2 and 3-dimensional orbifolds in semisimple Lie groups, and we focus on computing their dimension. For hyperbolic 3-orbifolds, we consider the component of the variety of characters that contains the holonomy composed with the principal representation, we show that its dimension equals half the dimension of the variety of characters of the boundary. We also show that this is a lower bound for the dimension of generic components. We furthermore provide tools for computing dimensions of varieties of characters of 2-orbifolds, including the Hitchin component. We apply this computation to the dimension growth of varieties of characters of some 3-dimensional manifolds in SL(n,C).

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arXiv2020

Geometry of the SL(3,C)-character variety of torus knots

Let G be the fundamental group of the complement of the torus knot of type (m,n). This has a presentation G= . We find the geometric description of the character variety X(G) of characters of representations of G into SL(3,C), GL(3,C) and PGL(3,C).

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arXiv2019

Asymptotics of twisted Alexander polynomials and hyperbolic volume

For a hyperbolic knot and a natural number n, we consider the Alexander polynomial twisted by the n-th symmetric power of a lift of the holonomy. We establish the asymptotic behavior of these twisted Alexander polynomials evaluated at unit complex numbers, yielding the volume of the knot exterior. More generally, we prove the asymptotic behavior for cusped hyperbolic manifolds of finite volume. The proof relies on results of Müller, and Menal-Ferrer and the last author. Using the uniformity of the convergence, we also deduce a similar asymptotic result for the Mahler measures of those polynomials.

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arXiv2019

Actions on products of CAT(-1) spaces

We show that for $X$ a proper $\mathrm{CAT}(-1)$ space there is a maximal open subset of the horofunction compactification of $X\times X$ with respect to the maximum metric that compactifies the diagonal action of an infinite quasi-convex group of the isometries of $X$. We also consider the product action of two quasi-convex representations of an infinite hyperbolic group on the product of two different proper $\mathrm{CAT}(-1)$ spaces.

math.GT↗
arXiv2019

Examples of character varieties in characteristic $p$ and ramification

We study $\mathrm{SL}_2(\mathbb{F})$-character varieties of knots over algebraically closed fields $\mathbb{F}$. We give a sufficient condition in terms of the double branched cover of a $2$-bridge knot (or, equivalently, of its Alexander polynomial) on the characteristic of $\mathbb{F}$, an odd prime, for the $\mathrm{SL}_2(\mathbb{F})$-character variety to present ramification phenomena. Finally we provide several explicit computations of character varieties to illustrate the result, exhibiting also other types of ramification.

math.GT↗
arXiv2018

Volumes of $\mathrm{SL}_n\mathbb{C}$-representations of hyperbolic 3-manifolds

Let $M$ be a compact oriented three-manifold whose interior is hyperbolic of finite volume. We prove a variation formula for the volume on the variety of representations of $M$ in $\operatorname{SL}_n(\mathbb C)$. Our proof follows the strategy of Reznikov's rigidity when $M$ is closed, in particular we use Fuks' approach to variations by means of Lie algebra cohomology. When $n=2$, we get back Hodgson's formula for variation of volume on the space of hyperbolic Dehn fillings. Our formula also yields the variation of volume on the space of decorated triangulations obtained by Bergeron-Falbel-Guillou and Dimofte-Gabella-Goncharov.

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