1,721,073 research outputs found
Maximal index automorphisms of free groups with no attracting fixed points on the boundary are Dehn twists.
No full textIn this paper we define a quantity called the rank of an outer automorphism of a free group which is the same as the index introduced in [D. Gaboriau, A. Jaeger, G. Levitt and M. Lustig, 'An index for counting fixed points for automorphisms of free groups', Duke Math. J. 93 (1998) 425-452] without the count of fixed points on the boundary. We proceed to analyze outer automorphisms of maximal rank and obtain results analogous to those in [D.J. Collins and E. Turner, 'An automorphism of a free group of finite rank with maximal rank fixed point subgroup fixes a primitive element', J. Pure and Applied Algebra 88 (1993) 43-49]. We also deduce that all such outer automorphisms can be represented by Dehn twists, thus proving the converse to a result in [M.M. Cohen and M. Lustig, 'The conjugacy problem for Dehn twist automorphisms of free groups', Comment Math. Helv. 74 (1999) 179-200], and indicate a solution to the conjugacy problem when such automorphisms are given in terms of images of a basis, thus providing a moderate extension to the main theorem of Cohen and Lustig by somewhat different methods
La nascita delle intendenze. Problemi dell'amministrazione periferica nel regno di Napoli
No full textCentralisers of linear growth automorphisms of free groups
No full textIn this note we investigate the centraliser of a linearly growing element of Out(Fn) (that is, a root of a Dehn twist automorphism), and show that it has a finite index subgroup mapping onto a direct product of certain "equivariant McCool groups" with kernel a finitely generated free abelian group. In particular, this allows us to show it is VF and hence finitely presented
Metric properties of outer space
No full textWe define metrics on Culler-Vogtmann space, which are an analogue of the Teichmuller metric and are constructed using stretching factors. In fact the metrics we study are related, one being a symmetrised version of the other. We investigate the basic properties of these metrics, showing the advantages and pathologies of both choices. We show how to compute stretching factors between marked metric graphs in an easy way and we discuss the behaviour of stretching factors under iterations of automorphisms. We study metric properties of folding paths, showing that they are geodesic for the non-symmetric metric and, if they do not enter the thin part of Outer space, quasi-geodesic for the symmetric metri
Tra legislatori e interpreti. Saggio di storia delle idee giuridiche in italia meridionale.
No full textLe corti supreme nell'ottocento italiano. Prassi, formazione e controllo della magistratura
No full textDisplacements of automorphisms of free groups II: Connectivity of level sets and decision problems
No full textThis 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 and its simplicial bordification. We develop a theory for both reducible and irreducible autormorphisms. As we reach the bordification of 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. Here, by the ``level set" we intend to indicate the set of points displaced by \textit{at most } some amount, rather than exactly some amount; this is sometimes called a ``sub-level set". As an application, this result provides a stopping procedure for brute force search algorithms in . 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~\cite{FMlevelset}We decided to split that paper following the recommendations of a referee
Stretching factors, metrics and train tracks for free products
No full textIn 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.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 (in particular, answering to some questions raised in Axis in outer space (2011) concerning the axis bundle of irreducible automorphisms).Finally, 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
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