arXiv subjects

Renxing Wan

Publications and source records attributed to Renxing Wan.

8 recordsLinked to original sources

arXiv2026

Genericity of hyperbolic 3-manifolds via Dehn surgery

A significant result by Lickorish and Wallace shows that every closed, orientable 3-manifold can be obtained from a Dehn surgery on one link in 3-sphere. As links and Dehn surgeries vary vastly in the universe, a question arises: how can we describe their properties in vague? We introduce a counting model on links and Dehn surgeries, and prove that under this model, (1) a randon link is hyperbolic; (2) a random 3-manifold is hyperbolic.

math.GT↗

arXiv2026

Tits Alternative in groups with proper product actions on proper Gromov-hyperbolic spaces

In this paper, we study groups with property (PPH), i.e., there exist finitely many proper Gromov-hyperbolic spaces $X_1,\ldots, X_l$ on which $G$ acts cocompactly such that the diagonal action of $G$ on the $\ell^1$-product $\prod_{i=1}^lX_i$ is proper. We show that any finitely generated subgroup of a finitely generated group with property (PPH) either is amenable or contains $F_2$. Furthermore, we study groups with property (PPT), i.e., groups with property (PPH) so that $X_1,\cdots,X_l$ are all proper quasi-trees. We show that any finitely generated subgroup of a finitely generated group with property (PPT) either is virtually (locally-finite)-by-$\mathbb{Z}^n$ or contains $F_2$. Additionally, we establish that for a non-elementary hyperbolic group $G$, $G$ admits a proper diagonal action on a finite product of regular trees if and only if $G$ has property (PPT). This result transforms a question posed by Button \cite{But19} into the problem of whether every non-elementary hyperbolic group has property (PPT).

math.GR↗

arXiv2025

Central extensions and proper actions on products of hyperbolic spaces

The main result of this paper identifies boundedness of the Euler class as the exact obstruction to preserving property QT under central extensions. For a central extension of groups $1\to Z\to E\to G\to 1$, we prove that $E$ has property QT if and only if $Z$ is finitely generated, $G$ has property QT, and the Euler class of the extension is bounded. This is achieved by using quasimorphisms as a bridge between central extensions and group actions. As applications, we show that mapping class groups of finite-type surfaces possibly with boundary, multicurve stabilizers, and outer automorphism groups of torsion-free one-ended hyperbolic groups have property QT. We also show that Sela's central extension description of the latter has a bounded Euler class. In addition, we introduce property PH, which is a weaker analogue of property QT related to locally uniform exponential growth of groups, and derive the same stability results under central extensions. We provide several examples with or without property PH. In particular, the fundamental group of a compact orientable $3$-manifold $M$ has property PH whenever no summand in the sphere-disk decomposition of $M$ supports Nil geometry.

math.GR↗

arXiv2025

A general construction of simultaneously hyperbolic elements

In this paper, we give an explicit construction of simultaneously hyperbolic elements in a group acting on finitely many Gromov-hyperbolic spaces under the weakest conditions. This essentially generalizes results of Clay-Uyanik in \cite{CU18}, of Genevois in \cite{Gen19}, and of Balasubramanya-Fern\'{o}s in \cite{BF24}. Besides, we show that the set of simultaneously hyperbolic elements has strictly positive density with respect to any proper word metric under the weakest conditions. This recovers many classical counting results, eg. the main result of Wiest in \cite{Wie17}. As an important ingredient in the proof of main results, we show that the set of simultaneously contracting elements in a group acting on finitely many metric spaces with contracting property has strictly positive density with respect to any proper word metric. This generalizes two results of Wan-Xu-Yang in \cite{WXY24} and of Balasubramanya-Fern\'{o}s in \cite{BF24}.

math.GR↗

arXiv2025

Quasi-retracts of groups

In this paper, we study a special class of quasi-homomorphisms, i.e. quasi-retractions from a group to its subgroups. We first give some algebraic and geometric properties of quasi-retracts and then propose a theory of quasi-split short exact sequences of groups. Later, we establish a connection between quasi-homomorphisms and induced quasi-actions. Finally, we give some geometric applications of quasi-homomorphisms, including normal quasi-retracts inherit cobounded actions on hyperbolic spaces, properties (QFA), (QT') and (PH') are all stable under left quasi-split group extensions, quasi-isomorphic groups have isomorphic hyperbolic structures, and so on.

math.GR↗

arXiv2024

Relative bounded cohomology on groups with contracting elements

Let $G$ be a countable group acting properly on a metric space with contracting elements and $\{H_i:1\le i\le n\}$ be a finite collection of Morse subgroups in $G$. We prove that each $H_i$ has infinite index in $G$ if and only if the relative second bounded cohomology $H^{2}_b(G, \{H_i\}_{i=1}^n; \mathbb{R})$ is infinite-dimensional. In addition, we also prove that for any contracting element $g$, there exists $k>0$ such that $H^{2}_b(G, \langle \langle g^k\rangle \rangle; \mathbb{R})$ is infinite-dimensional. Our results generalize a theorem of Pagliantini-Rolli for finite-rank free groups and yield new results on the (relative) second bounded cohomology of groups.

math.GR↗

arXiv2024

Marked length spectrum rigidity in groups with contracting elements

This paper presents a study of the well-known marked length spectrum rigidity problem in the coarse-geometric setting. For any two (possibly non-proper) group actions $G\curvearrowright X_1$ and $G\curvearrowright X_2$ with contracting property, we prove that if the two actions have the same marked length spectrum, then the orbit map $Go_1\to Go_2$ must be a rough isometry. In the special case of cusp-uniform actions, the rough isometry can be extended to the entire space. This generalizes the existing results in hyperbolic groups and relatively hyperbolic groups. In addition, we prove a finer marked length spectrum rigidity from confined subgroups and further, geometrically dense subgroups. Our proof is based on the Extension Lemma and uses purely elementary metric geometry. This study produces new results and recovers existing ones for many more interesting groups through a unified and elementary approach.

math.GR↗

arXiv2023

Uniform exponential growth for groups with proper product actions on hyperbolic spaces

This paper studies the locally uniform exponential growth and product set growth for a finitely generated group $G$ acting properly on a finite product of hyperbolic spaces. Under the assumption of coarsely dense orbits or shadowing property on factors, we prove that any finitely generated non-virtually abelian subgroup has uniform exponential growth. These assumptions are fulfilled in many hierarchically hyperbolic groups, including mapping class groups, specially cubulated groups and BMW groups. Moreover, if $G$ acts weakly acylindrically on each factor, we show that, with two exceptional classes of subgroups, $G$ has uniform product set growth. As corollaries, this gives a complete classification of subgroups with product set growth for any group acting discretely on a simply connected manifold with pinched negative curvature, for groups acting acylindrically on trees, and for 3-manifold groups.

math.GR↗