Buildings, projective geometry - what led Tits to think of "the field with one element"?

Question

The mysterious object "field with one element" seems to appear first in J. Tits papers on buildings. It is mentioned in almost any text on $\mathbb{F}_1$. However, I have never seen any exposition of his ideas understandable by pedestrians, beyond something very short remarks (see example below).

Question: Can one give some comment on buildings, their relations to projective geometries and how that might lead to idea of a field with one element? And what intuition should we get from that for $\mathbb{F}_1$?

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For example Connes, Consani "On the notion of geometry over $F_1$" wrote the following. I am not an expert and that does not give me a clear picture.

In his theory of buildings J. Tits obtained a broad generalization of the celebrated von Staudt reconstruction theorem in projective geometry, involving as groups of symmetries not only GLn but the full collection of Chevalley algebraic groups. Among the axioms ([28]) which characterize these constructions, a relevant one is played by the condition of “thickness” which states, in its simplest form, that a projective line contains at least three points. By replacing this requirement with its strong negation, i.e. by imposing that a line contains exactly two points, one still obtains a coherent “geometry” which is a degenerate form of classical projective geometry. In the case of buildings, this degenerate case is described by the theory of “thin” complexes and in particular by the structure of the apartments, which are the basic constituents of the theory of buildings. The degeneracy of the von Staudt field inspired to Tits the conviction that these degenerate forms of geometries are a manifestation of the existence of a hypothetical algebraic object that he named “the field of characteristic one” ([26]). The richness and beauty of this geometric picture gives convincing evidence for the pertinence of a separate study of the degenerate case.

Answer 1

I've found some info related to my own question and so let me share it.

That is how $F_1$ is related to projective geometry (I cannot say on buildings yet).

• Projective spaces can be described by axioms Projective plane and higher dimensional projective spaces can be described via axioms, very simple ones, roughly speaking, saying that for any two points there is one line, for any line there are two points etc (see e.g. Wikipedia).

• Axioms are satisfied by the degenerate example of set with n+1 points There is a degenerate example of such projective plane consisting of just 3 points. With obvious definition of lines it satisfies main axioms of the projective place except last one non-degeneracy axiom. And similar for higher dimensional projective spaces - one can take n+1 poins define lines, planes, k-planes, hyperplaces in an obvious way - just the subsets consisiting of k-points - and again it will satisfy natural axioms for projective space.

• Agrees with the limit q->1 Consider the projective space of dimension n, over finite field F_q. It has $[n+1]_q$ points, sending q->1 we get n+1 points on "F_1" projective space, it agrees with previos item. Moreover counting lines,planes, k-planes in $P^n(F_q)$ and sending q->1 we get the same result as counting of subplanes in the previous item.

• Symmetric group acts by "projective transformations" Considering the group of "projective transformations" (i.e. those which preserves lines) we will get the symmetric group as the group of projective transformations. Actually condition "preserve lines" is empty.

Thus axiomatic view on projective geometry somehow leads to intuition that there should be F_1, such that projective space is set of n+1 points, and S_n = GL(F_1).

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Now some PUZZLE.

It is expected that there should be some algebraic extension of F_1, the "fields" $F_{1^k}$. Similar to $F_{q^k}$.

Question What are projective spaces for $F_{1^k}$ ?

Answer (imho) we do not get nonthing new - we get the same sets with n+1 points as $P^n$. It seems it is somewhat conter-intuitive. But I do know see other options.

Arguments:

• It is proposed in Kapranov, Smirnov "Cohomology determinants and reciprocity laws" page 4 (see also MO272698) that GL(n,F_{1^k}) is group of monomial matrices with roots of unit of degree k as elements. So the group has k^n n! elements. $P^{n} = GL(n+1)/(GL(n)\times GL(1))$ from this formula we get n+1 element for arbitrary "k" - dependence on "k" disappears.

• It can be proved for from the axioms of projective plane that any plane has q^2+q+1 points for some "q" (for non-degenerate planes) or seven examples of degenerate planes. For non-degenerate case it is proved that there is no plane for q=6 or q=10 etc, (no examples known for $q\ne p^m$). So there is no way to match "k" and "q", for non-degenerate planes, and for degenerated ones - they are not symmetric with respect to the group action (except 3 point plane). So only 3-point plane can be a reasonable candidate indendently on "k".

Answer 2

for a general reference, this book "Absolute Arithmetic and $\mathbb{F}_1$-Geometry" (Thas, ed., with chapters by Borger, Deitmar, Le Bruyn, Lorscheid, Manin & Marcolli, Thas) should give all the answers, I guess. Some chapters might be on arxiv.

And I also think this paper ''Projective spaces over $\mathbb{F}_{1^\ell}$'' by Thas (on arxiv) might give the answers to the question(s) of @Alexander Chervov.

Answer 3

A good survey seems to be found in this paper by Lopez Pena and Lorscheid.