[Paper Review] Controlled Floyd Separation and Non Relatively Hyperbolic Groups

Shubhabrata Das, Mahan Mj|arXiv (Cornell University)|Aug 5, 2014

Geometric and Algebraic Topology17 references4 citations

TL;DR

This paper constructs a finitely generated group $G = F_2 *_K F_2$, where $K$ is an infinitely generated malnormal subgroup of a free group, such that $G$ is not hyperbolic relative to any proper subgroups, has uncountably many geodesic rays Floyd-separated with respect to quasigeodesics, and possesses a trivial Floyd boundary despite strong geometric features like being a direct limit of hyperbolic CAT(0) cubulated groups and having all finitely presented subgroups hyperbolic.

ABSTRACT

We introduce the notion of controlled Floyd separation between geodesic rays starting at the identity in a finitely generated group G. Two such geodesic rays are said to be Floyd separated with respect to quasigeodesics if the (Floyd) length of c-quasigeodesics (for fixed but arbitrary c) joining points on the geodesic rays is asymptotically bounded away from zero. This is always satisfied by Morse geodesics. The main purpose of this paper is to furnish an example of a finitely generated group $G$ such that 1) all finitely presented subgroups of G are hyperbolic, 2) G has an uncountable family of geodesic rays that are Floyd separated with respect to quasigeodesics, 3) G is not hyperbolic relative to any collection of proper subgroups. 4) G is a direct limit of hyperbolic CAT(0) cubulated groups. 5) G has trivial Floyd boundary in the usual sense. On the way towards constructing $G$, we construct a malnormal infinitely generated (and hence non-quasiconvex) subgroup of a free group, giving negative evidence towards a question of Swarup and Gitik.

Motivation & Objective

• To construct a finitely generated group $G$ that is not hyperbolic relative to any proper subgroups, despite having strong geometric and algebraic features.
• To demonstrate the existence of uncountably many geodesic rays in $G$ that are Floyd-separated with respect to quasigeodesics, indicating non-trivial boundary structure under controlled metrics.
• To show that the Floyd boundary of $G$ is trivial (a single point), challenging known obstructions to triviality such as the presence of $\mathbb{Z} \oplus \mathbb{Z}$ or wide groups.
• To provide a counterexample to potential characterizations of non-trivial Floyd boundaries by constructing a group with controlled Floyd separation but trivial boundary.
• To establish the existence of an infinitely generated malnormal subgroup in a free group, offering negative evidence to a question by Swarup and Gitik on quasiconvexity and malnormality.

Proposed method

• Construct $G = F_2 *_K F_2$ as a double of a free group along an ascending union $K = \bigcup_n K_n$ of malnormal quasiconvex subgroups.
• Use graphical graded small cancellation theory to ensure $G$ satisfies $GSC_3$ and is lacunary hyperbolic, implying hyperbolicity in the asymptotic cone.
• Establish controlled Floyd separation by analyzing quasigeodesic paths between geodesic rays in $G$, showing their Floyd lengths are bounded away from zero.
• Prove triviality of the Floyd boundary by showing all limit points of subgroups $F_2(a,b)$ and $F_2(x,y)$ coincide under the Floyd metric, using relators $R_m$ with symmetric quasigeodesic structure.
• Use asymptotic cones and tree-graded space theory to analyze the geometry of $G$, showing that limit circles intersect trivially and cut points do not exist.
• Apply results from relative hyperbolicity, quasiconvexity, and convergence actions to show that the action on the Floyd boundary is minimal and trivial.

Experimental results

Research questions

• RQ1 Can a finitely generated group have uncountably many geodesic rays that are Floyd-separated with respect to quasigeodesics while being non-relatively hyperbolic?
• RQ2 Is the triviality of the Floyd boundary sufficient to imply relative hyperbolicity, or can it occur in groups with strong geometric features like direct limits of hyperbolic cubulated groups?
• RQ3 Can an infinitely generated malnormal subgroup exist in a free group, and what does this imply for the quasiconvexity and hyperbolic embedding of such subgroups?
• RQ4 Does controlled Floyd separation between geodesic rays imply non-triviality of the Floyd boundary, or can such separation coexist with trivial boundary?
• RQ5 Can a group be a direct limit of hyperbolic cubulated groups and still have a trivial Floyd boundary, despite all finitely presented subgroups being hyperbolic?

Key findings

• The group $G = F_2 *_K F_2$ is not hyperbolic relative to any collection of proper subgroups, as shown in Theorem 4.1.
• There exist uncountably many geodesic rays in $G$ that are Floyd-separated with respect to $(\kappa,\kappa)$-quasigeodesics for any $\kappa > 1$, as established in Theorem 7.1.
• The Floyd boundary $\partial_f G$ consists of a single point, making it trivial, as proven in Theorem 8.1.
• All finitely presented subgroups of $G$ are hyperbolic, and $G$ contains no Baumslag-Solitar groups $BS(m,n)$, as stated in Proposition 3.11.
• The subgroup $K = \bigcup_n K_n$ is malnormal and infinitely generated in $F_2$, providing a negative answer to a question of Swarup and Gitik regarding quasiconvexity of malnormal subgroups.
• The asymptotic cone of $G$ contains no cut points, and limit circles intersect the asymptotic cone of $K_2$ trivially, as shown in Theorem 6.7.