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Stefano Francaviglia

Publications and source records attributed to Stefano Francaviglia.

At least 19 recordsLinked to original sources
arXiv2024

On the action of relatively irreducible automorphisms on their train tracks

Let $G$ be a group and let ${\mathcal G}$ be a free factor system of $G$, namely a free splitting of $G$ as $G=G_1*\dots*G_k*F_r$. In this paper, we study the set of train track points for ${\mathcal G}$-irreducible automorphisms $ϕ$ with exponential growth (relatively to ${\mathcal G}$). Such set is known to coincide with the minimally displaced set $\operatorname{Min}(ϕ)$ of $ϕ$. Our main result is that $\operatorname{Min}(ϕ)$ is co-compact, under the action of the cyclic subgroup generated by $ϕ$. Along the way we obtain other results that could be of independent interest. For instance, we prove that any point of $\operatorname{Min}(ϕ)$ is in uniform distance from $\operatorname{Min}(ϕ^{-1})$. We also prove that the action of $G$ on the product of the attracting and the repelling trees for $ϕ$, is discrete. Finally, we get some fine insight about the local topology of relative outer space. As an application, we generalise a classical result of Bestvina, Feighn and Handel for the centralisers of irreducible automorphisms of free groups, in the more general context of relatively irreducible automorphisms of a free product. We also deduce that centralisers of elements of $\operatorname{Out}(F_3)$ are finitely generated, which was previously unknown. Finally, we mention that an immediate corollary of co-compactness is that $\operatorname{Min}(ϕ)$ is quasi-isometric to a line.

math.GR↗
arXiv2023

The Conjugacy Problem for $Out(F_3)$

We present a solution to the Conjugacy Problem in the group of outer-automorphisms of $F_3$, a free group of rank 3. We distinguish according to several computable invariants, such as irreducibility, subgroups of polynomial growth, and subgroups carrying the attracting lamination. We establish, by considerations on train tracks, that the conjugacy problem is decidable for the outer-automorphisms of $F_3$ that preserve a given rank 2 free factor. Then we establish, by consideration on mapping tori, that it is decidable for outer-automorphisms of $F_3$ whose maximal polynomial growth subgroups are cyclic. This covers all the cases left by the state of the art.

math.GR↗
arXiv2023

Bounded Cohomology Classes of Exact Forms

On negatively curved compact manifolds, it is possible to associate to every closed form a bounded cocycle - hence a bounded cohomology class - via integration over straight simplices. The kernel of this map is contained in the space of exact forms. We show that in degree 2 this kernel is trivial, in contrast with higher degree. In other words, exact non-zero $2$-forms define non-trivial bounded cohomology classes. This result is the higher dimensional version of a classical theorem by Barge and Ghys for surfaces. As a consequence, one gets that the second bounded cohomology of negatively curved manifolds contains an infinite dimensional space, whose classes are explicitly described by integration of forms. This also showcases that some recent results by Marasco (arXiv:2202.04419, arXiv:2209.00560) can be applied in higher dimension to obtain new non-trivial results on the vanishing of certain cup products and Massey products. Some other applications are discussed.

math.GT↗
arXiv2022

A transfer principle: from periods to isoperiodic foliations

We classify the possible closures of leaves of the isoperiodic foliation (sometimes called absolute period foliation) defined on the Hodge bundle, i.e. the moduli space of abelian differentials over genus $g\geq 2$ smooth curves, and prove that the foliation is ergodic on those sets. The results derive from the connectedness properties of the fibers of the period map defined on the Torelli cover of the moduli space. Some consequences on the topology of Hurwitz spaces of primitive branched coverings over elliptic curves are also obtained. To prove the results we develop the theory of augmented Torelli space, the branched Torelli cover of the Deligne-Mumford compactification of the moduli space of curves.

math.AG↗
arXiv2021

An easy (horizontal) walk through fake octagons

A fake octagon is a genus two translation surface with only one singular point and the same periods as the octagon. Existence of infinitely many fakes was first established by McMullen in 2007, and more generally follows from dynamical properties of the so called isoperiodic foliation. The purpose of this note is to build an explicit infinite family of fake octagons, constructed by means of elementary methods. We describe an easy cut-and-paste surgery and show that the all iterates of that surgery produce fake octagons. We prove that any iterate gives a fake which is different from each other, and we show that any such fake can be approximated arbitrarily well by some other fake of the family. This note is intended to be elementary and fully accessible to non-expert readers.

math.DS↗
arXiv2021

Displacements of automorphisms of free groups II: Connectivity of level sets and decision problems

This is the second of two papers in which we investigate the properties of displacement functions of automorphisms of free groups (more generally, free products) on the Culler-Vogtmann Outer space $CV_n$ and its simplicial bordification. We develop a theory for both reducible and irreducible autormorphisms. As we reach the bordification of $CV_n$ we have to deal with general deformation spaces, for this reason we developed the theory in such generality. In first paper~\cite{FMpartI} we studied general properties of the displacement functions, such as well-orderability of the spectrum and the topological characterization of min-points via partial train tracks (possibly at infinity). This paper is devoted to proving that for any automorphism (reducible or not) any level set of the displacement function is connected. As an application, this result provides a stopping procedure for brute force search algorithms in $CV_n$. We use this to reprove two known algorithmic results: the conjugacy problem for irreducible automorphisms and detecting irreducibility of automorphisms. Note: the two papers were originally packed together in the preprint arxiv:1703.09945. We decided to split that paper following the recommendations of a referee.

math.GR↗
arXiv2020

Displacements of automorphisms of free groups I: Displacement functions, minpoints and train tracks

This is the first of two papers in which we investigate the properties of the displacement functions of automorphisms of free groups (more generally, free products) on Culler-Vogtmann Outer space and its simplicial bordification - the free splitting complex - with respect to the Lipschitz metric. The theory for irreducible automorphisms being well-developed, we concentrate on the reducible case. Since we deal with the bordification, we develop all the needed tools in the more general setting of deformation spaces, and their associated free splitting complexes. In the present paper we study the local properties of the displacement function. In particular, we study its convexity properties and the behaviour at bordification points, by geometrically characterising its continuity-points. We prove that the global-simplex-displacement spectrum of $Aut(F_n)$ is a well-ordered subset of $\mathbb R$, this being helpful for algorithmic purposes. We introduce a weaker notion of train tracks, which we call {\em partial train tracks} (which coincides with the usual one for irreducible automorphisms) and we prove that, for any automorphism, points of minimal displacement - minpoints - coincide with the marked metric graphs that support partial train tracks. We show that any automorphism, reducible or not, has a partial train track (hence a minpoint) either in the outer space or its bordification. We show that, given an automorphism, any of its invariant free factors is seen in a partial train track map. In a subsequent paper we will prove that level sets of the displacement functions are connected, and we will apply that result to solve certain decision problems.

math.GR↗
arXiv2020

The minimally displaced set of an irreducible automorphism is locally finite

We prove that the minimally displaced set of a relatively irreducible automorphism of a free splitting, situated in a deformation space, is uniformly locally finite. The minimally displaced set coincides with the train track points for an irreducible automorphism. We develop the theory in a general setting of deformation spaces of free products, having in mind the study of the action of reducible automorphisms of a free group on the simplicial bordification of Outer Space. For instance, a reducible automorphism will have invariant free factors, act on the corresponding stratum of the bordification, and in that deformation space it may be irreducible (sometimes this is referred as relative irreducibility).

math.GR↗
arXiv2020

Local deformations of branched projective structures: Schiffer variations and the Teichmüller map

We study a class of continuous deformations of branched complex projective structures on closed surfaces of genus $g\geq 2$, which preserve the holonomy representation of the structure and the order of the branch points. In the case of non-elementary holonomy we show that when the underlying complex structure is infinitesimally preserved the branch points are necessarily arranged on a canonical divisor, and we establish a partial converse for hyperelliptic structures.

math.CV↗
arXiv2020

The mimimally displaced set of an irreducible automorphism of $F_N$ is co-compact

We study the minimally displaced set of irreducible automorphisms of a free group. Our main result is the co-compactness of the minimally displaced set of an irreducible automorphism with exponential growth $ϕ$, under the action of the centraliser $C(ϕ)$. As a corollary, we get that the same holds for the action of $<ϕ>$ on $Min(ϕ)$. Finally, we prove that the minimally displaced set of an irreducible automorphism of growth rate one is consisted of a single point.

math.GR↗
arXiv2018

Volume rigidity at ideal points of the character variety of hyperbolic 3-manifolds

Given the fundamental group $Γ$ of a finite-volume complete hyperbolic $3$-manifold $M$, it is possible to associate to any representation $ρ:Γ\rightarrow \text{Isom}(\mathbb{H}^3)$ a numerical invariant called volume. This invariant is bounded by the hyperbolic volume of $M$ and satisfies a rigidity condition: if the volume of $ρ$ is maximal, then $ρ$ must be conjugated to the holonomy of the hyperbolic structure of $M$. This paper generalizes this rigidity result by showing that if a sequence of representations of $Γ$ into $\text{Isom}(\mathbb{H}^3)$ satisfies $\lim_{n \to \infty} \text{Vol}(ρ_n) = \text{Vol}(M)$, then there must exist a sequence of elements $g_n \in \text{Isom}(\mathbb{H}^3)$ such that the representations $g_n \circ ρ_n \circ g_n^{-1}$ converge to the holonomy of $M$. In particular if the sequence $ρ_n$ converges to an ideal point of the character variety, then the sequence of volumes must stay away from the maximum. We conclude by generalizing the result to the case of $k$-manifolds and representations in $\text{Isom}(\mathbb H^m)$, where $m\geq k$.

math.GT↗
arXiv2017

On the connectivity of level sets of automorphisms of free groups, with applications to decision problems

We show that the level sets of automorphisms of free groups with respect to the Lipschitz metric are connected as subsets of Culler-Vogtmann space. In fact we prove our result in a more general setting of deformation spaces. As applications, we give metric solutions of the conjugacy problem for irreducible automorphisms and the detection of reducibility. We additionally prove technical results that may be of independent interest --- such as the fact that the set of displacements is well ordered.

math.GR↗
arXiv2015

Stretching factors, metrics and train tracks for free products

In this paper we develop the metric theory for the outer space of a free product of groups. This generalizes the theory of the outer space of a free group, and includes its relative versions. The outer space of a free product is made of $G$-trees with possibly non-trivial vertex stabilisers. The strategies are the same as in the classical case, with some technicalities arising from the presence of infinite-valence vertices. In particular, we describe the Lipschitz metric and show how to compute it; we prove the existence of optimal maps; we describe geodesics represented by folding paths. We show that train tracks representative of irreducible (hence hyperbolic) automorphisms exist and that their are metrically characterized as minimal displaced points, showing in particular that the set of train tracks is closed. We include a proof of the existence of simplicial train tracks map without using Perron-Frobenius theory. A direct corollary of this general viewpoint is an easy proof that relative train track maps exist in both the free group and free product case.

math.GR↗
arXiv2014

The oriented graph of multi-graftings in the Fuchsian case

We prove the connectedness and calculate the diameter of the oriented graph of graftings associated to exotic complex projective structures on a compact surface S with a given holonomy representation of Fuchsian type. The oriented graph of graftings is the graph whose vertices are the equivalence classes of marked CP^1-structures on S with a given fixed holonomy, and there is an oriented edge between two structures if the second is obtained from the first by grafting.

math.GT↗
arXiv2013

Spectral rigidity of automorphic orbits in free groups

It is well-known that a point $T\in cv_N$ in the (unprojectivized) Culler-Vogtmann Outer space $cv_N$ is uniquely determined by its \emph{translation length function} $||.||_T:F_N\to\mathbb R$. A subset $S$ of a free group $F_N$ is called \emph{spectrally rigid} if, whenever $T,T'\in cv_N$ are such that $||g||_T=||g||_{T'}$ for every $g\in S$ then $T=T'$ in $cv_N$. By contrast to the similar questions for the Teichmüller space, it is known that for $N\ge 2$ there does not exist a finite spectrally rigid subset of $F_N$. In this paper we prove that for $N\ge 3$ if $H\le Aut(F_N)$ is a subgroup that projects to an infinite normal subgroup in $Out(F_N)$ then the $H$-orbit of an arbitrary nontrivial element $g\in F_N$ is spectrally rigid. We also establish a similar statement for $F_2=F(a,b)$, provided that $g\in F_2$ is not conjugate to a power of $[a,b]$. We also include an appended corrigendum which gives a corrected proof of Lemma 5.1 about the existence of a fully irreducible element in an infinite normal subgroup of of $Out(F_N)$. Our original proof of Lemma 5.1 relied on a subgroup classification result of Handel-Mosher, originally stated by Handel-Mosher for arbitrary subgroups $H\le Out(F_N)$. After our paper was published, it turned out that the proof of the Handel-Mosher subgroup classification theorem needs the assumption that $H$ be finitely generated. The corrigendum provides an alternative proof of Lemma~5.1 which uses the corrected, finitely generated, version of the Handel-Mosher theorem and relies on the 0-acylindricity of the action of $Out(F_N)$ on the free factor complex (due to Bestvina-Mann-Reynolds). A proof of 0-acylindricity is included in the corrigendum.

math.GR↗
arXiv2012

Stable complexity and simplicial volume of manifolds

Let the complexity of a closed manifold M be the minimal number of simplices in a triangulation of M. Such a quantity is clearly submultiplicative with respect to finite coverings, and by taking the infimum on all finite coverings of M normalized by the covering degree we can promote it to a multiplicative invariant, a characteristic number already considered by Milnor and Thurston, which call the "stable complexity" of M. We study here the relation between the stable complexity of M and Gromov's simplicial volume ||M||. It is immediate to show that ||M|| is smaller or equal than the stable complexity of M and it is natural to ask whether the two quantities coincide on aspherical manifolds with residually finite fundamental group. We show that this is not always the case: there is a constant C_n<1 such that ||M|| is smaller than C_n times the stable complexity for any hyperbolic manifold M of dimension at least 4. The question in dimension 3 is still open in general. We prove that the stable complexity equals ||M|| for any aspherical irreducible 3-manifold M whose JSJ decomposition consists of Seifert pieces and/or hyperbolic pieces commensurable with the figure-eight knot complement. The equality holds for all closed hyperbolic 3-manifolds if a particular three-dimensional version of the Ehrenpreis conjecture is true.

math.GT↗
arXiv2012

Branched projective structures with quasi-Fuchsian holonomy

We prove that if S is a closed compact surface of negative Euler characteristic, and if R is a quasi-Fuchsian representation in PSL(2,C), then the deformation space M(k,R) of branched projective structures on S with total branching order k and holonomy R is connected, as soon as k>0. Equivalently, two branched projective structures with the same quasi-Fuchsian holonomy and the same number of branch points are related by a movement of branch points. In particular grafting annuli are obtained by moving branch points. In the appendix we give an explicit atlas for the space M(k,R). It is shown to be a smooth complex manifold modeled on Hurwitz spaces.

math.GT↗