Gromov boundary

The Cayley graph of a free group with two generators. This is a hyperbolic group whose Gromov boundary is a Cantor set. Hyperbolic groups and their boundaries are important topics in geometric group theory, as are Cayley graphs.

The (6,4,2) triangular hyperbolic tiling. The triangle group corresponding to this tiling has a circle as its Gromov boundary.

In mathematics, the Gromov boundary of a delta-hyperbolic space (especially a hyperbolic group) is an abstract concept generalizing the boundary sphere of hyperbolic space. Conceptually, the Gromov boundary is the set of all points at infinity. For instance, the Gromov boundary of the real line is two points, corresponding to positive and negative infinity.

Definition

There are several equivalent definitions of the Gromov boundary. One of the most common uses equivalence classes of geodesic rays.

Pick some point $O$ of a hyperbolic metric space $X$ to be the origin. A geodesic ray is a path given by an isometry $\gamma :[0,\infty )\rightarrow X$ such that each segment $\gamma ([0,t])$ is a path of shortest length from $O$ to $\gamma (t)$.

We say that two geodesics $\gamma _{1},\gamma _{2}$ are equivalent if there is a constant $K$ such that $d(\gamma _{1}(t),\gamma _{2}(t))\leq K$ for all $t$. The equivalence class of $\gamma$ is denoted $[\gamma ]$.

The Gromov boundary of a hyperbolic metric space $X$ is the set $\partial X=\{[\gamma ]|\gamma$ is a geodesic ray in $X\}$.

Topology

It is useful to use the Gromov product of three points. The Gromov product of three points $x,y,z$ in a metric space is $(x,y)_{z}=1/2(d(x,z)+d(y,z)-d(x,y))$. In a tree (graph theory), this measures how long the paths from $z$ to $x$ and $y$ stay together before diverging. Since hyperbolic spaces are tree-like, the Gromov product measures how long geodesics from $z$ to $x$ and $y$ stay close before diverging.

Given a point $p>$ in the Gromov boundary, we define the sets $V(p,r)=\{q\in \partial X|$ there are geodesic rays $\gamma _{1},\gamma _{2}$ with $[\gamma _{1}]=p,[\gamma _{2}]=q$ and $\lim \inf _{s,t\rightarrow \infty }(\gamma _{1}(s),\gamma _{2}(t))_{O}\geq r\}$. These open sets form a basis for the topology of the Gromov boundary.

These open sets are just the set of geodesic rays which follow one fixed geodesic ray up to a distance $r$ before diverging.

Properties of the Gromov boundary

The Gromov boundary has several important properties:

• The Gromov boundary of a hyperbolic group is compact.
• If a group $G$ acts geometrically on a delta-hyperbolic space, then $G$ is hyperbolic group and $G$ and $X$ have homeomorphic Gromov boundaries.
• The Gromov boundary of a tree is a Cantor set.
• The number of ends of a hyperbolic group is the number of components of the Gromov boundary.

Cannon's Conjecture

Main article: Cannon's conjecture

Cannon's conjecture concerns the classification of groups with a 2-sphere at infinity:

Cannon's conjecture: Every Gromov hyperbolic group with a 2-sphere at infinity acts geometrically on hyperbolic 3-space.

The analog to this conjecture is known to be true for 1-spheres and false for spheres of all dimension greater than 2.

References

• Kapovich, Ilya, and Nadia Benakli. "Boundaries of hyperbolic groups." Combinatorial and geometric group theory (New York, 2000/Hoboken, NJ, 2001) 296 (2002): 39-93.