arXiv subjects

Theodore Weisman

Publications and source records attributed to Theodore Weisman.

13 recordsLinked to original sources

arXiv2026

Stability for boundary actions of cocompact lattices in Euclidean buildings

When $X$ is a locally compact Euclidean building, the isometry group of $X$ acts by homeomorphisms on the space of $k$-simplices in the visual boundary of $X$. We consider perturbations of these actions for discrete groups of isometries acting with compact quotient on $X$, showing that all small enough perturbations are semi-conjugate to the original action. This proves in particular that, when $Q$ is any parabolic subgroup in a semisimple $p$-adic Lie group $G$, the induced action of a cocompact lattice in $G$ has a topologically stable action on $G/Q$.

math.DS↗

arXiv2025

Dehn filling in semisimple Lie groups

We generalize one part of Thurston's hyperbolic Dehn filling theorem to arbitrary-rank semisimple Lie groups by showing that certain deformations of extended geometrically finite subgroups of a semisimple Lie group are still extended geometrically finite. As a special case, our theorem gives a criterion which guarantees that a deformation of a relatively Anosov subgroup is (non-relatively) Anosov, and also ensures that limit sets vary continuously. Our result also applies to several higher-rank examples in convex projective geometry which are outside of the relatively Anosov setting.

math.GT↗

arXiv2024

Singular value gap estimates for free products of semigroups

We establish lower estimates for singular value gaps of free products of $1$-divergent semigroups $\Gamma_1,\Gamma_2\subset \mathsf{GL}_d(\mathbb{K})$ which are in ping-pong position. As an application, we prove that if $\Gamma_1$ and $\Gamma_2$ are quasi-isometrically embedded subgroups in ping pong position, then the group they generate $\langle \Gamma_1,\Gamma_2\rangle$ is also quasi-isometrically embedded. In addition, we establish that the class of linear finitely generated groups, admitting a faithful linear representation over $\mathbb{R}$ which is a quasi-isometric embedding, is closed under free products.

math.GR↗

arXiv2024

Limits of limit sets in rank-one symmetric spaces

We consider the question of continuity of limit sets for sequences of geometrically finite subgroups of isometry groups of rank-one symmetric spaces, and prove analogues of classical (Kleinian) theorems in this context. In particular we show that, assuming strong convergence of the sequence of subgroups, the limit sets vary continuously with respect to Hausdorff distance, and if the sequence is weakly type-preserving, the sequence of Cannon-Thurston maps also converges uniformly to a limiting Cannon-Thurston map. Our approach uses the theory of extended geometrically finite representations, developed recently by the second author.

math.GT↗

arXiv2024

Morse properties in convex projective geometry

We study properties of "hyperbolic directions" in groups acting cocompactly on properly convex domains in real projective space, from three different perspectives simultaneously: the (coarse) metric geometry of the Hilbert metric, the projective geometry of the boundary of the domain, and the singular value gaps of projective automorphisms. We describe the relationship between different definitions of "Morse" and "regular" quasi-geodesics arising in these three different contexts. This generalizes several results of Benoist and Guichard to the non-Gromov hyperbolic setting.

math.GT↗

arXiv2024

Topological stability of relatively hyperbolic groups acting on their boundaries

We prove a topological stability result for the actions of hyperbolic groups on their Bowditch boundaries. More precisely, we show that a sufficiently small perturbation of the standard boundary action, if assumed on each parabolic subgroup to be a perturbation by semi-conjugacy, is in fact always globally semi-conjugate to the standard action. This proves a relative version of the main result of arXiv:2206.14914. The assumption of control on the perturbation of parabolics is necessary.

math.GR↗

arXiv2023

Examples of extended geometrically finite representations

This is the second of a pair of papers on extended geometrically finite (EGF) representations, which were originally posted as a single article under the title "An extended definition of Anosov representation for relatively hyperbolic groups." In this paper, we prove that the holonomy representation of a projectively convex cocompact manifold with relatively hyperbolic fundamental group is always an EGF representation. We also prove that EGF representations arise as holonomy representations of convex projective manifolds with generalized cusps and as compositions of projectively convex cocompact representations with symmetric representations of SL(d, R). We additionally show that any small deformation of a representation of the latter form is still EGF.

math.GT↗

arXiv2023

Cubulated hyperbolic groups admit Anosov representations

We prove that any hyperbolic group acting properly discontinuously and cocompactly on a $\mathrm{CAT}(0)$ cube complex admits a projective Anosov representation into $\mathrm{SL}(d, \mathbb{R})$ for some $d$. More specifically, we show that if $\Gamma$ is a hyperbolic quasiconvex subgroup of a right-angled Coxeter group $C$, then a generic representation of $C$ by reflections restricts to a projective Anosov representation of $\Gamma$.

math.GR↗

arXiv2023

Stability of hyperbolic groups acting on their boundaries

A hyperbolic group acts by homeomorphisms on its Gromov boundary. We use a dynamical coding of boundary points to show that such actions are topologically stable in the dynamical sense: any nearby action is semi-conjugate to (and an extension of) the standard boundary action. This result was previously known in the special case that the boundary is a topological sphere. Our proof here is independent and gives additional information about the semiconjugacy in that case. Our techniques also give a new proof of global stability when the boundary is a circle.

math.GR↗

arXiv2023

Dynamical properties of convex cocompact actions in projective space

We give a dynamical characterization of convex cocompact group actions on properly convex domains in projective space in the sense of Danciger-Gueritaud-Kassel: we show that convex cocompactness in $\mathbb{R} \mathrm{P}^d$ is equivalent to an expansion property of the group about its limit set, occuring in different Grassmannians. As an application, we give a sufficient and necessary condition for convex cocompactness for groups which are hyperbolic relative to a collection of convex cocompact subgroups. We show that convex cocompactness in this situation is equivalent to the existence of an equivariant homeomorphism from the Bowditch boundary to the quotient of the limit set of the group by the limit sets of its peripheral subgroups.

math.GT↗

arXiv2023

Combination theorems for geometrically finite convergence groups

We prove combination theorems in the spirit of Klein and Maskit in the context of discrete convergence groups acting geometrically finitely on their limit sets. As special cases, we obtain combination theorems for geometrically finite groups of isometries of Hadamard manifolds with pinched negative curvature, and for relatively quasi-convex subgroups of relatively hyperbolic groups.

math.GR↗

arXiv2022

An extended definition of Anosov representation for relatively hyperbolic groups

We define a new family of discrete representations of relatively hyperbolic groups which unifies many existing definitions and examples of geometrically finite behavior in higher rank. The definition includes the relative Anosov representations defined by Kapovich-Leeb and Zhu, and Zhu-Zimmer, as well as holonomy representations of various different types of "geometrically finite" convex projective manifolds. We prove that these representations are all stable under deformations whose restriction to the peripheral subgroups satisfies a dynamical condition, in particular allowing for deformations which do not preserve the conjugacy class of the peripheral subgroups.

math.GR↗

arXiv2017

Realization of groups with pairing as Jacobians of finite graphs

We study which groups with pairing can occur as the Jacobian of a finite graph. We provide explicit constructions of graphs whose Jacobian realizes a large fraction of odd groups with a given pairing. Conditional on the generalized Riemann hypothesis, these constructions yield all groups with pairing of odd order, and unconditionally, they yield all groups with pairing whose prime factors are sufficiently large. For groups with pairing of even order, we provide a partial answer to this question, for a certain restricted class of pairings. Finally, we explore which finite abelian groups occur as the Jacobian of a simple graph. There exist infinite families of finite abelian groups that do not occur as the Jacobians of simple graphs.

math.CO↗