soft geometric structure emerging from a rigid axiomatic system

[José Manuel Rodríguez Caballero]

Dear FOM members,
  I would like to share the following opinion due to M. Gromov [page 6, 1]:

One believe nowadays that most essential invariants of an infinite group
> [denoted Gamma] are quasi-isometry invariants.
>

Gromov attributes the germ of this idea to Mostow's rigidity theorem [2],
which states that the geometry of a complete, finite-volume hyperbolic
manifold of dimension at least 3 is determined by its fundamental group.

For Gromov, the structure of a group (as a metric space with respect to the
word metric) is

rigid, crystalline
>

whereas the structure of its asymptotic space (how an observer infinitely
far away from the identity sees the group) is

a soft and flabby chunk of geometry
>

We could imagine the structure of an axiomatic system as rigid and
crystalline. Also, we can define a kind of asymptotic space of this
axiomatic system by generalizing Gromov's construction. This generalization
could be achieved, following Chaitin's approach [3] for descriptive
complexity, by defining the distance between two statements X and Y in this
axiomatic system as the maximum between of the minimum of the number of
steps needed to prove X assuming Y and minimum of the number of steps
needed to prove Y assuming X. To prove X given X is considered that it
takes zero steps. We tolerate infinite distances. Finally, concentrate on
the asymptotic properties which are expressible in terms of the distances
between variable statements in this axiomatic system as these distances go
to infinity.

Are there references to the "soft and flabby chunk of geometry" that may
emerge from the metric space defined on an axiomatic system in the way
indicated above?

Kind regards,
Jose M.


Reference:
[1] Gromov, Mikhail. "Geometric group theory, Vol. 2: Asymptotic invariants
of infinite groups." (1996).
URL = https://www.ihes.fr/~gromov/metricinvariants/126/

[2] Mostow, G. D. (1968), "Quasi-conformal mappings in n-space and the
rigidity of the hyperbolic space forms", Publ. Math. IHES, 34: 53–104
URL = http://www.numdam.org/article/PMIHES_1968__34__53_0.pdf

[3] Chaitin, Gregory J. "Information-theoretic incompleteness." *Applied
Mathematics and Computation* 52.1 (1992): 83-101.
URL = https://www.sciencedirect.com/science/article/abs/pii/009630039290099M
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