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

Abstract: 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 $\ell<sup>1$-product $\prod_{i=1}<sup>lX_i$ 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-$\mathbb{Z}<sup>n$ 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).

• The paper establishes that for groups with proper product actions on Gromov-hyperbolic spaces, every finitely generated subgroup is either amenable or contains a free subgroup F₂.
• The paper demonstrates that for groups with actions on quasi-tree factors, each subgroup is either virtually (locally finite)-by-ℤⁿ or contains F₂, refining subgroup classifications.
• The paper proves an equivalence for non-elementary hyperbolic groups between proper diagonal actions on products of regular trees and property (PPT), linking geometric dynamics with algebraic structure.

Tits Alternative for Groups with Proper Product Actions on Proper Gromov-Hyperbolic Spaces

Introduction and Main Results

This paper investigates geometric and algebraic properties of finitely generated groups acting properly on finite products of proper Gromov-hyperbolic spaces, focusing on the Tits Alternative for subgroups. Two distinct group-theoretic properties are central: (PPH) (proper product action on proper Gromov-hyperbolic spaces) and (PPT) (proper product action on proper quasi-trees, with each factor space quasi-isometric to a tree). The authors establish the following principal dichotomies:

1. For groups with (PPH): Every finitely generated subgroup is either amenable or contains $F_2$.
2. For groups with (PPT): Every finitely generated subgroup is either virtually (locally finite)-by-$\mathbb{Z}^n$ or contains $F_2$.

A further equivalence is established for non-elementary hyperbolic groups $G$: The existence of a proper diagonal action on a finite product of regular trees is shown to be equivalent to $G$ having property (PPT).

Technical Foundations

The arguments exploit the geometry of Gromov-hyperbolic spaces and the dynamical classification of group actions thereon: elliptic, parabolic (horocyclic), lineal, focal, and general type, as in the sense of Gromov [Gro87] and recent syntheses [ABO19]. Key tools include:

• Product actions: The notion of the diagonal action of $G$ on the $\ell^1$-product $\prod_{i=1}^l X_i$, with each $X_i$ admitting an isometric, cocompact $G$-action.
• Busemann quasimorphisms: The paper uses results from [CCMT15] connecting the nature of isometric group actions on hyperbolic spaces to the structure of homogeneous quasimorphisms, paired with amenable group properties.
• Classification techniques and regular focal actions: The characterization of regular focal actions is used to analyze group structure in relation to product actions.

Weak and Strong Tits Alternatives

(PPH) Context

The first main theorem demonstrates that for any finitely generated group $\mathbb{Z}^n$0 with (PPH), every finitely generated subgroup is either amenable or contains a non-abelian free subgroup $\mathbb{Z}^n$1. This extends the Tits Alternative beyond linear and CAT(0) cube complex settings, and encompasses direct products of hyperbolic groups, certain solvable Baumslag–Solitar groups, and other natural examples.

Amenability here is deduced via a careful analysis of possible action types:

• If the subgroup admits only bounded, horocyclic, or lineal actions, its algebraic structure is strongly constrained.
• If there is any component with a general type action, one can always extract $\mathbb{Z}^n$2.

Moreover, precise structural statements are proved, including that, in absence of horocyclic actions, a subgroup is virtually abelian of rank at most $\mathbb{Z}^n$3 precisely when all actions are elliptic or lineal, while the existence of a general type action is exactly when $\mathbb{Z}^n$4 is present.

(PPT) Context and Detailed Structure Description

The second main result addresses the case where the $\mathbb{Z}^n$5 are quasi-trees (i.e., spaces quasi-isometric to trees). Here, every finitely generated subgroup is either virtually (locally finite)-by-$\mathbb{Z}^n$6 or contains $\mathbb{Z}^n$7. This sharpens the dichotomy: certain torsion phenomena, including infinite locally finite subgroups, must carry through explicitly in the subgroup structure. This also incorporates lamplighter groups and finite direct products of free groups as key examples—both classes act properly on products of quasi-trees.

Additionally, if a finitely generated group with property (PPT) has no infinite locally finite subgroup, then every finitely generated subgroup is virtually abelian or contains $\mathbb{Z}^n$8 (a classical Tits Alternative).

The proof techniques show that for virtually (locally finite)-by-$\mathbb{Z}^n$9 groups, the torsion comes from locally finite normal subgroups, and the quotient is (at most) finitely generated abelian.

Connection to Proper Actions on Products of Trees

A highlight is the detailed equivalence for hyperbolic groups:

A non-elementary hyperbolic group admits a proper diagonal action on a finite product of regular trees if and only if it has property (PPT).

This is established by combining quasi-action rigidity results of [But23] and metric rigidity properties. It translates a question of Button into a purely group-theoretic statement: Does every non-elementary hyperbolic group have property (PPT)?

The analysis rules out certain action types (notably horocyclic and lineal) in this context, showing that any proper product action restricts to general type actions on each regular tree factor. The methods rely on properties of group stabilizers, absence of infinite torsion subgroups for hyperbolic groups, and structure of minimal invariant subtrees.

Amenability, Locally Finite Groups, and Property (QT)

The treatment of amenable groups yields the following claim:

• A finitely generated amenable group with no infinite locally finite subgroup has property (QT) if and only if it is virtually abelian.

This result, generalizing earlier theorems about actions on CAT(0) cube complexes [CS11, SW05], demonstrates that virtually abelian structure is (essentially) the only way for an amenable group to have a proper quasi-tree product action without pathological torsion.

The paper provides explicit examples contrasting torsion phenomena: for instance, infinite locally finite groups can act properly on locally finite trees but do not have (PPH) or (PPT).

Examples and Applications

The authors analyze the Baumslag–Solitar groups $F_2$0, showing they have (PPH) for all $F_2$1, and for $F_2$2 these groups exhibit proper diagonal actions which are focal on both hyperbolic and tree factors.

The lamplighter group $F_2$3 is shown to have property (PPT), constructed via explicit ascending HNN structure and Bass–Serre theory, with proper diagonal action on a product of two regular trees.

Implications and Future Directions

This work systematizes Tits Alternative-type structure for subgroups of groups acting properly on direct products of Gromov-hyperbolic spaces, raising several natural open directions:

• For which (non-elementary) hyperbolic groups does property (PPT) hold, and thus is a proper action on a finite product of regular trees always available? Is there a hyperbolic group lacking this property?
• What is the precise relationship between geometric properties of spaces the group acts on (e.g., presence of quasi-trees, regular trees, CAT(0) cube complexes) and fine-grained algebraic structure?

The analysis deepens the understanding of the algebraic consequences of proper product actions in nonpositive curvature and hyperbolic settings, contextualizing Tits Alternative phenomena in a vastly larger class of groups and setting the stage for future work on rigidity and subgroup classification in this context.

Conclusion

This paper rigorously establishes Tits Alternative structures for groups with proper diagonal actions on finite products of proper Gromov-hyperbolic spaces and quasi-trees, yielding sharp subgroup dichotomies, subclassifying amenable groups, and connecting group action properties with the geometry of the underlying spaces. The results clarify the landscape of proper group actions in geometric group theory and provide a platform for further investigation into rigidity and cohomological properties of such groups (2604.04007).