Horofunctions and symbolic dynamics on Gromov hyperbolic groups. (Q2777598)

In hyperbolic space \(\mathbb{H}^n\), the family of horospheres tangent at a fixed point \(x\) at infinity is naturally described by a function \(h\colon\mathbb{H}^n\to\mathbb{R}\) from hyperbolic space \(\mathbb{H}^n\) to the real line \(\mathbb{R}\) having the following properties: (1) the point preimages \(h^{-1}(t)\) are the horospheres tangent to \(\partial(\mathbb{H}^n)\) at \(x\), (2) \(|h(y)- h(z)|\) measures the distance between the horospheres containing \(y\) and \(z\) and (3) \(h(y)\to-\infty\) as \(y\to x\). Such a function is called a Busemann function. There is a gradient geodesic flow perpendicular to the horospheres of \(h\) toward the point \(x\) at infinity.NEWLINENEWLINE If \(\mathbb{H}^n\) is replaced by the Cayley graph \(X\) of a Gromov hyperbolic group having space \(\partial(X)\) at infinity, then there are analogous functions, which the authors call `integral horofunctions'. The point preimages might be appropriate called (combinatorial) `horospheres'. Each horofunction maps naturally to a point at infinity, though many may map to the same point at infinity.NEWLINENEWLINE Horofunctions that differ by a constant are said to belong to the same class. The authors topologize the set \(\Phi_0\) of equivalence classes. Each class has a natural point at infinity. The resulting map \(\pi\colon\Phi_0\to\partial(X)\) is surjective, continuous, and uniformly finite to one.NEWLINENEWLINE The authors define a combinatorial gradient flow perpendicular to the horospheres. This induces a map \(\alpha\colon\Phi_0\to\Phi_0\) which in some sense shows how horospheres change as one flows toward infinity.NEWLINENEWLINE The main result is this: Theorem. The dynamical system \((\Phi_0,\alpha)\) is topologically conjugate to a subshift of finite type.NEWLINENEWLINE The authors will use the ideas from this paper elsewhere to obtain a symbolic coding for the geodesic flow associated to a Gromov-hyperbolic group.