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When speaking about hyperbolic groups/spaces, one usually refers to Gromov's monograph Hyperbolic groups for their introduction. However, coarse notions of hyperbolicity can be found in some of his earlier texts, such that Infinite groups as geometric objects and Hyperbolic manifolds, groups and action. Also, among the (many) equivalent definitions of $\delta$-hyperbolic spaces, one of them refers to Rips' condition, suggesting that E. Rips played some role here.

Hence my question:

What is the prehistory of Gromov-hyperbolic spaces/groups?

Here, "prehistory" refers to "before Gromov's monograph".

Edit: I should specify that I am not looking for the historical motivations of the definition, but the origins of the definition itself. How did the definition emerge?

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    $\begingroup$ There’s always Dehn’s algorithm and small cancellation theory, which was a primary motivator for investigating and defining hyperbolic groups. Is this the kind of thing you are looking for, or are you more interested in direct geometric statements? $\endgroup$
    – Carl-Fredrik Nyberg Brodda
    Commented Jun 6, 2021 at 7:08
  • $\begingroup$ @Carl-FredrikNybergBrodda: In fact, I am not interested in general historical motivations, but more specifically in the evolution of the notion of coarse hyperbolicity before Gromov's monograph. $\endgroup$
    – AGenevois
    Commented Jun 6, 2021 at 11:20

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Gromov, in his monograph (see page 83 of "Essays in group theory"), writes the following.

The idea of hyperbolicity has been lingering in combinatorial group theory since the basic work by Dehn. An extensive study of a class of word hyperbolic groups $\Gamma$ with $\mathrm{dim}\, \partial \Gamma = 1$ (in the combinatorial disguise) was conducted by Olshanski (see [Ol]). Deep algebraic results on general hyperbolic groups are contained in the as yet unpublished work by I. Rips who calls them groups with negative curvature.

As you note, Gromov's earlier papers contain relevant work. Gromov also refers to Cannon's work, but only as regards rationality of certain languages. I believe that Cannon's treatment of quasi-geodesic stability, and its relevance for hyperbolic groups, slightly predates Gromov's.

Both Cannon and Gromov are alive and both reply to email. So perhaps it would be better to contact them directly.

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    $\begingroup$ It is also worth mentioning that it was this paper of Cannon that led directly to Thurston's definition of automatic groups. Thurston observed that the conditions that Cannon was proving, such as finite number of cone types, and fellow-travelling could be formulated in terms of finite state automata. $\endgroup$
    – Derek Holt
    Commented Jun 6, 2021 at 7:49
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    $\begingroup$ Eliyahu Rips is also still alive by the way. $\endgroup$
    – Loreno Heer
    Commented Jun 11, 2021 at 10:02
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    $\begingroup$ @LorenoHeer As is A. Yu. Olshanskii. $\endgroup$
    – Carl-Fredrik Nyberg Brodda
    Commented Jun 11, 2021 at 21:04
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The notion of coarse curvature was very popular in the "Leningrad math school" where Gromov is from before Gromov, for example, see the works of D. Alexandrov and Toponogov. The small cancelation idea which goes back to Dehn was very popular in other parts of the USSR, especially in Moscow (Grindlinger, Novikov, Adian, Olshanskii and others) but did not influence Gromov's work very much, although groups with thin geodesic triangles are studied already in the work by Novikov and Adian. So in some sense Alexandrov, Toponogov, Novikov and Adian predated Gromov.

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    $\begingroup$ Greendlinger was in Tula, was he not? $\endgroup$
    – Carl-Fredrik Nyberg Brodda
    Commented Jun 11, 2021 at 20:58
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    $\begingroup$ Greendlinger worked mostly in Tula (also in New York, Moscow, Ivanovo, Philadelphia, etc.), but he often visited and gave talks in Moscow and communicated closely with Moscow mathematicians, so he may be considered a member of the Moscow school too. $\endgroup$
    – markvs
    Commented Jun 11, 2021 at 21:10
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    $\begingroup$ Correct: Grindlinger $\endgroup$
    – markvs
    Commented Jun 14, 2021 at 13:34
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    $\begingroup$ Well, he was born Greendlinger in the US -- Grindlinger is just the English transliteration of the Russian transliteration of the name. $\endgroup$
    – Carl-Fredrik Nyberg Brodda
    Commented Jun 14, 2021 at 15:17

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