arXiv research

We consider horofunction compactifications of symmetric spaces with respect to invariant Finsler metrics. We show that any (generalized) Satake compactification can be realized as a horofunction compactification with respect to a polyhedral Finsler metric.

The paper studies horofunction compactifications of symmetric cones under Finsler distances.

We show that the horofunction compactification of Teichmüller space with the Teichmüller metric is homeomorphic to the Gardiner-Masur compactification.

Extends Teichmüller distance concept to non-distance maps.

Study shows horofunction compactification's topology matches dual norm's unit ball.

We establish a natural and geometric 1-1 correspondence between projective toric varieties of dimension $n$ and horofunction compactifications of $\mathbb{R}^n$ with respect to rational polyhedral norms. For this purpose, we explain a topological model of toric varieties. Consequently, toric varieties in algebraic geom…

The horoboundary of Teichmüller space is path connected and has non-dense Busemann points.

The arc metric is an asymmetric metric on the Teichm{ü}ller space T(S) of a surface S with nonempty boundary. In this paper we study the relation between Thurston's compactification and the horofunction compactification of T(S) endowed with the arc metric. We prove that there is a natural homeomorphism between the two …

In this paper we answer positively a question raised by Kapovich and Leeb in a paper titled "Finsler bordifications of symmetric and certain locally symmetric spaces". Specifically, we show that for a finite-dimensional vector space with a polyhedral norm, its horofunction compactification is homeomorphic to the dual u…

Researchers describe horofunctions in noncompact Hermitian symmetric spaces.

In this work we describe horofunction compactifications of metric spaces and finite dimensional real vector spaces through asymmetric metrics and asymmetric polyhedral norms by means of nonstandard methods, that is, ultrapowers of the spaces at hand. The polyhedral compactifications of the vector spaces carry the struc…

Optimal geodesics connect boundary points in Teichmüller space.

We give an example of a horocycle in the Teichmüller space of the five-times-punctured sphere that does not converge in the Gardiner--Masur compactification, or equivalently in the horofunction compactification of the Teichmüller metric. As an intermediate step, we exhibit a simple closed curve whose extremal length is…

Paper develops a theory for Patterson-Sullivan measures in higher rank symmetric spaces.

Using the identification of the symmetric space $\mathrm{SL}(n,\mathbb{R})/\mathrm{SO}(n)$ with the Teichmüller space of flat $n$-tori of unit volume, we explore several metrics and compactifications of these spaces, drawing inspiration both from Teichmüller theory and symmetric spaces. We define and study analogs of t…

We determine the asymptotic behaviour of extremal length along arbitrary Teichmüller rays. This allows us to calculate the endpoint in the Gardiner-Masur boundary of any Teichmüller ray. We give a proof that this compactification is the same as the horofunction compactification. An important subset of the latter is the…

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-…

We give a geometric interpretation of the maximal Satake compactification of symmetric spaces $X=G/K$ of noncompact type, showing that it arises by attaching the horofunction boundary for a suitable $G$-invariant Finsler metric on $X$. As an application, we establish the existence of natural bordifications, as orbifold…

We determine the set of Busemann points of an arbitrary finite-dimensional normed space. These are the points of the horofunction boundary that are the limits of "almost-geodesics". We prove that all points in the horofunction boundary are Busemann points if and only if the set of extreme sets of the dual unit ball is …

New metric on geodesic currents connects different surface genera.

We fully describe the horofunction boundary $\partial_h L_2$ with the word metric associated with the generating set $\{t,at\}$ (i.e the metric arising in the Diestel-Leader graph $\text{DL}(2,2)$). The visual boundary $\partial_\infty L_2$ with this metric is a subset of $\partial_h L_2$. Although $\partial_\infty L_2…

This paper studies convergence of horospheres in CAT(0) spaces.

We introduce the functor * which assigns to every metric space X its symmetric join *X. As a set, *X is a union of intervals connecting ordered pairs of points in X. Topologically, *X is a natural quotient of the usual join of X with itself. We define an Isom(X)-invariant metric d* on *X. Classical concepts known for H…

We show that the horofunction boundary of Teichmüller space with Thurston's Lipschitz metric is the same as the Thurston boundary. We use this to determine the isometry group of the Lipschitz metric, apart from in some exceptional cases. We also show that the Teichmüller spaces of different surfaces, when endowed with …

Study boundary actions on CAT(0) spaces, proving topological freeness.

We construct a triangulation of a compactification of the Moduli space of a surface with at least one puncture that is closely related to the Deligne-Mumford compactification. Specifically, there is a surjective map from the compactification we construct to the Deligne-Mumford compactification so that the inverse image…

Extends harmonic maps compactification to punctured Riemann surfaces.

Characterizes toroidal and semi-toric compactifications as log minimal models and applies to weak K-moduli.

Satake has constructed compactifications of symmetric spaces D=G/K which (under a condition called geometric rationality by Casselman) yield compactifications of the corresponding locally symmetric spaces. The different compactifications depend on the choice of a representation of G. One example is the Baily-Borel-Sata…

The paper studies compactifications of SL(2,C) character varieties for punctured surfaces.

New coordinates for Teichmüller space compactification.

New compactification for character varieties with good topological properties.

Embeds Higson compactification into adelic solenoids.

We discuss the `hd-compactification' of a semi-simple Lie group to a manifold with corners; it is the real analog of the wonderful compactification of deConcini and Procesi. There is a 1-1 correspondence between the boundary faces of the compactification and conjugacy classes of parabolic subgroups with the boundary fa…

We define a compactification of symmetric spaces of noncompact type, seen as spaces of isometry classes of marked lattices, analogous to the Thurston compactification of the Teichmüller space, and we show that it is equivariantly isomorphic to a Satake compactification. We then use it to define a new compactification o…

We construct several examples of compactifications of Einstein metrics. We show that the Eguchi--Hanson instanton admits a projective compactification which is non--metric, and that a metric cone over any (pseudo)--Riemannian manifolds admits a metric projective compactification. We construct a para--$c$--projective co…

Researchers create a new compactification of character varieties using geometric and algebraic methods.

Paper studies compactifications of Higgs bundles and self-duality equations.

No natural topological compactification for Fulton-MacPherson.

Paper shows geometric properties preserved by compactifications in relation to coarse structures and group actions.

The paper classifies and computes limits of equivariant compactifications of groups.

The study finds only finitely many Kähler-Einstein compactifications for semisimple groups.

The paper extends end concepts to arbitrary groups and spaces.

In this paper we present a topological way of building a compactification of a symmetric space from a compactification of a Weyl Chamber.

To study a noncompact Riemannian manifold, it is often useful to find a compactification. We discuss several common compactifications and survey some recent results.

The group of direct isometries of the real n-dimensional hyperbolic space is G=SOo(n,1). This isometric action admits many differentiable compactifications into an action on the closed ball. We prove that all such compactifications are topologically conjugate but not necessarily differentiably conjugate. We give the cl…

In arXiv:1503.08402v2 Gelander described a new compactification of the moduli space of finite area hyperbolic surfaces using invariant random subgroups. The goal of this paper is to relate this compactification to the classical augmented moduli space, also known as the Deligne-Mumford compactification. We define a cont…

Consider a finite dimensional (generally reducible) polynomial representation ρof GL_n. A projective compactification of GL_n is the closure of ρ(GL_n) in the space of all operators defined up to a factor (this class of spaces can be characterized as equivariant projective normal compactifications of GL_n). We give an …