arXiv subjects
Publications and source records attributed to Xenia Flamm.
We develop the theory of $\Theta$-positive representations from general Fuchsian groups to linear groups over real closed fields. Our definition, which does not assume the boundary map to be continuous, encompasses many generalizations of positive or Anosov representations that have been considered in the literature.
Positive tuples of complete flags in $\mathbb{R}^3$ define two convex polygons in $\mathbb{RP}^2$, one inscribed in the other. We are interested in relating the Holmes-Thompson area of the inner polygon for the Hilbert metric on the outer polygon to the double and triple ratios of the positive tuple of flags. This article focuses on positive triples and quadruples of flags. For quadruples, we investigate the special cases of hyperbolic quadrilaterals and the parametrization of the finite area convex real projective structures on a thrice-punctured sphere.
We define a notion of morphism for generalized affine buildings, also known as affine $\Lambda$-buildings, extending existing definitions and giving rise to a category of generalized affine buildings. For affine $\Lambda$-buildings equipped with a transitive group action, we provide sufficient conditions for the existence of morphisms between them. As an application, we investigate under which conditions morphisms or isomorphisms between various generalized affine buildings from the literature (defined via lattices, norms, non-standard symmetric spaces, or \`a la Bruhat-Tits) can be defined. For generalized affine buildings coming from non-standard symmetric spaces we further show functoriality for subgroups and under change of valued field.
We develop a theory of Hilbert geometry over general ordered valued fields, associating with an open convex subset of the projective space a quotient Hilbert metric space. Under natural non-degeneracy assumptions, we prove that the ultralimit of a sequence of rescaled real Hilbert geometries is isometric to the Hilbert metric space of an open convex projective subset over a Robinson field. This result allows us to prove that ideal points of the space of convex real projective structures on a closed manifold arise from actions on non-Archimedean Hilbert geometries without global fixed point. We explicitly describe the Hilbert metric space of a non-Archimedean bounded polytope $P$ defined over a subfield of the valuation ring as the geometric realization of the flag complex of $P$ modeled on a Weyl chamber. As an application, we obtain a complete description of Gromov-Hausdorff limits of a real polytope with rescaled Hilbert metric.
Let $p \colon Y \to X$ be a finite, regular cover of finite graphs with associated deck group $G$, and consider the first homology $H_1(Y;\mathbb{C})$ of the cover as a $G$-representation. The main contribution of this article is to broaden the correspondence and dictionary between the representation theory of the deck group $G$ on the one hand, and topological properties of homology classes in $H_1(Y;\mathbb{C})$ on the other hand. We do so by studying certain subrepresentations in the $G$-representation $H_1(Y;\mathbb{C})$. The homology class of a lift of a primitive element in $π_1(X)$ spans an induced subrepresentation in $H_1(Y;\mathbb{C})$, and we show that this property is never sufficient to characterize such homology classes if $G$ is Abelian. We study $H_1^{\textrm{comm}}(Y;\mathbb{C}) \leq H_1(Y;\mathbb{C})$ -- the subrepresentation spanned by homology classes of lifts of commutators of primitive elements in $π_1(X)$. Concretely, we prove that the span of such a homology class is isomorphic to the quotient of two induced representations. Furthermore, we construct examples of finite covers with $H_1^{\textrm{comm}}(Y;\mathbb{C}) \neq \ker(p_*)$.
Let $\mathbb{F}$ be the field of real Puiseux series and $\mathcal{T}_\mathbb{F}$ the $\mathbb{Q}$-tree defined by Brumfiel. We show that completing all the segments of $\mathcal{T}_\mathbb{F}$ does not result in a complete metric space.
These notes transcribe a workshop about the notion of total positivity and $Θ$-positivity and its relation to Higher Teichmüller Theory. $Θ$-positivity is a notion of positivity in semisimple Lie groups and was recently introduced by Guichard and Wienhard as a generalization of Lusztig's total positivity. It is believed to be the cathartic notion to classify higher Teichmüller spaces. Without doubt, substantial progress will be achieved in the near future on the study of $Θ$-positive structures. These notes provide an account of the state of the art as of 2021.
The main result of this article is that Hitchin representations over real closed field extensions $\mathbb{F}$ of $\mathbb{R}$ correspond precisely to those representations of the fundamental group of a closed surface into $\textrm{PSL}(n,\mathbb{F})$ that are conjugate to $\mathbb{F}$-positive representations, i.e. representations that admit an equivariant limit map from the set of fixed points in the boundary of the universal cover of the surface into the set of full flags in $\mathbb{F}^n$ satisfying specific positivity properties. As the theorem treats general real closed fields, and not only the reals, the tools of analysis are not available. Instead, our proof is based on the Tarski-Seidenberg transfer principle and a multiplicative version of the Bonahon-Dreyer coordinates. We use this result to prove that $\mathbb{F}$-positive representations form semi-algebraically connected components of the space of all representations, that consist entirely of injective and discrete representations, which are positively hyperbolic and weakly dynamics-preserving over $\mathbb{F}$. Furthermore, we show how to associate intersection geodesic currents to $\mathbb{F}$-positive representations, and conclude with applications to the Weyl chamber length compactification and to dual spaces of geodesic currents.