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Comparing the Floyd and ideal boundaries of a metric space. (English) Zbl 1182.54030

Let $\left(X,d\right)$ be a metric space. In $X$ we can define different types of boundaries: the Ideal boundary of $X$ denoted by ${\partial }_{1}X$; the Gromov boundary, denoted by ${\partial }_{G}X$; the $g$-boundary, denoted by ${\partial }_{g}X$.

The Ideal boundary is defined using geodesic rays. The Gromov boundary is defined through Gromov sequences (sequences converging to infinity). The Gromov Boundary is usually defined only for Gromov hyperbolic spaces but the authors extended this concept to arbitrary metric spaces. The $g$-boundary is the boundary of an unbounded length space which replaces an unbounded metric $l$ by a bounded metric $\sigma$, using a conformal distortion involving a function $g$.

Actually spherical and Floyd boundaries are types of $g$-boundaries. In fact, let $g:\left[0,\infty \right)\to \left(0,\infty \right)$ be a measurable function such that for a constant $C>2$ one can establish the following conditions (where $G\left(t\right):={\int }_{0}^{\infty }g\left(s\right)\phantom{\rule{0.166667em}{0ex}}ds,t\ge 0$):

(F1) $g\left(t\right)\le Cg\left(s\right)$ when $s,t\ge 0$ and $s-1\le t\le 2s+1$,

(S1) $g\left(t\right)\le Cg\left(s\right)$ when $s,t\ge 0$ and $\left(s-1\right)/2\le t\le 2s+1$,

(F2) $G\left(0\right)\le Cg\left(0\right)$,

(S2) $G\left(t\right)\le C\left(1+t\right)g\left(t\right)$, $t\ge 0$,

$g$ is a weak $C$-Floyd function if conditions (F1) and (F2) are satisfied, $g$ is a Floyd function if (S1) and (F2) are satisfied and $g$ is a $C$-sphericalizing function if (S1) and (S2) are satisfied. The spherical boundary was a consequence of the concept of sphericalization introduced in by Z. Balogh and S. M. Buckley [Conform. Geom. Dyn. 9, 76–101, electronic only (2005; Zbl 1108.30033)]. The purpose was to interpret quasyhyperbolizations of unbounded length spaces in the context of unbounded space. In the article under review the authors aim to discuss $g$-boundaries comparing them with ideal and Gromov boundaries in order to emphasize both similarities and differences. Some of the results shown are also found in the literature available but the authors prove comparison results for larger classes of spaces $X$ and functions $g$. The article is organized in three sections. The first, notation and preliminaries, includes a section on CAT(0) and Gromov hyperbolic spaces and one on spherical and Floyd boundaries, for, as the authors say in the introduction, understanding the spherical boundary is vital for deciding when sphericalization is an invertible process, (for more details see the article mentioned above). The second section explains the main comparison theorems. It includes five comparison theorems with proof.

The authors include also one last section with an extensive list of counterexamples to show that the assumptions given in the comparison theorems are indeed needed.

MSC:
 54D35 Extensions of topological spaces (compactifications, supercompactifications, completions, etc.) 30F45 Conformal metrics (hyperbolic, Poincaré, distance functions) 53A30 Conformal differential geometry