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I have a couple of basic questions about visual hyperbolic metric spaces. A visual hyperbolic metric space $X$ is a hyperbolic metric space with the property that there exists an $o \in X$ such that any $x \in X$ is a distance $\leq K$ from a geodesic ray originating from $o$.

  1. If $X$ is the Cayley graph of a (non-elementary) word hyperbolic group equipped with the word distance as the metric, is $X$ visual?

  2. Are there simple examples of hyperbolic metric spaces that are not visual?

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    $\begingroup$ Consider any finite group... Did you forget some assumptions? $\endgroup$ Commented Jan 18 at 23:56
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    $\begingroup$ Part 1 is a duplicate of this question. As a further exercise, construct a complete unbounded hyperbolic geodesic metric space which contains no geodesic rays whatsoever. $\endgroup$ Commented Jan 19 at 1:45
  • $\begingroup$ Indeed, the first part of the question is specifically for non-elementary hyperbolic groups, will fix. $\endgroup$ Commented Jan 19 at 3:02

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  1. Cayley graphs of hyperbolic groups are visual in this sense. This follows from the finiteness of “cone types”. See e.g. “Word Processing in Groups”.

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  1. Take a ray $[0,\infty)$ with intervals of length $n$ attached at each integer (as a subset of $\mathbb{R}^2$, we can take $[0,\infty) \times \{0\} \cup_n [(n,0),(n,n)]$). This is a tree, so is $0$-hyperbolic, but it is not visual.
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