Several papers, including this and this claim that divergence of finitely generated groups and metric spaces have been introduced by Misha Gromov in his paper "Asymptotic invariants of infinite groups". What is the precise reference and how did Gromov call this invariant?
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6$\begingroup$ It is really annoying that some authors quote a result contained in a 194-pages long paper, without saying the precise place where it can be found. $\endgroup$– Francesco PolizziCommented Jun 25, 2018 at 13:27
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4$\begingroup$ It is especially true for Gromov's papers where it is not always immediately clear what he is talking about. The paradox is that after it is explained, one feels that the interpretation is very clear and the only reasonable one. $\endgroup$– user6976Commented Jun 25, 2018 at 13:38
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$\begingroup$ "A really good idea is one that is obvious, after someone else explains it to you" - Anonymous- :) $\endgroup$– Francesco PolizziCommented Jun 25, 2018 at 13:42
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$\begingroup$ In a quick look at the book I can't locate any allusion to divergence. Have you asked the authors of these linked papers? $\endgroup$– YCorCommented Jun 25, 2018 at 22:01
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Here is (what I believe is) the relevant paragraph of the second "this", of which I am one of the co-authors. This paragraph contains a reference to a particular passage from Gromov's Asymptotic Invariants paper including a citation to the particular paragraph in that paper where the relevant discussion occurs.
In symmetric spaces of non-compact type, the order of the divergence of geodesic rays is either exponential (when the rank is one) or linear (when the rank is at least two). This inspired an initial thought that in the presence of non-positive curvature the divergence must be either linear or exponential. See [Gro93] for a discussion—an explicit statement of this conjecture appears in 6.B2, subsection “Geometry of ∂T and Morse landscape at infinity,” Example (h). In particular, Gromov stated an expectation that all pairs of geodesic rays in the universal cover of a closed Riemannian manifold of non-positive curvature diverge either linearly or exponentially [Gro93].
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$\begingroup$ This does not answer my question, unfortunately. Gromov did not define divergence of groups or metric spaces in 6. $B_2$. I think the correct answer is "Gromov did not define divergence functions of groups or metric spaces, but a comment in 6.$B_2$ inspired Gersten to give a definition". $\endgroup$– user6976Commented Jun 26, 2018 at 7:33
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1$\begingroup$ In a sense this answers your question, which asks what is the part of Gromov's book that is aimed by these references. Based on this, you have the right to judge what is the part of Gromov's contribution to the definition of divergence (I'm curious what is the intended meaning that "two geodesic rays diverge exponentially"). $\endgroup$– YCorCommented Jun 26, 2018 at 11:28
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$\begingroup$ From personal experience, I would not be surprised that Gromov discussed this topic verbally with someone (Gersten?) to an unknown extent and precision. He may have even asked the specific question about the divergence function. When that person asked Gromov for a reference, he may have pointed to this paragraph, but the content of the discussion was probably richer than the content of the reference. $\endgroup$– ARGCommented Jun 26, 2018 at 11:37
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2$\begingroup$ I have asked Misha Gromov yesterday, and he said he did not define the divergence of groups or metric spaces in that book. $\endgroup$– user6976Commented Jun 26, 2018 at 12:35
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1$\begingroup$ As far as I understand, the divergence of two geodesic length-parametrised rays $r_1(t)$, $r_2(t)$ which Gromov discusses in 6.$B_2$ (where $r_1(0)=r_2(0)=o$) is the function $d(t)$ which is the distance between $r_1(t)$ and $r_2(t)$ in the space with open ball $B(o,t)$ removed. I think it is a standard notion in differential geometry (that is the meaning of the phrase that hyperbolic space-time is exponentially expanding). $\endgroup$– user6976Commented Jun 26, 2018 at 12:55