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The classical synthetic notion of projective plane consists of a set of points, a set of lines, and a relation of incidence between the two, such that any two distinct points lie on a unique line and any two distinct lines intersect in a unique point (plus some nondegeneracy assumptions). There are similar notions of projective 3-space, $n$-space, and so on — but 1-dimensional projective space seems harder to capture synthetically, since there is no "room", dimensionally, for subspaces in between the points and the entire space.

Has anyone attempted to define a synthetic notion of "projective line"? Ideally such a definition would have properties like the following:

  • The space $P^1(k)$ is naturally a projective line for any division ring $k$, and from any projective line $L$ satisfying enough axioms we can construct a skew field $c(L)$ such that $c(P^1(k)) \cong k$ and $P^1(c(L))\cong L$ (unnaturally). The corresponding facts for Desarguesian projective planes are classical.

  • Any line in a projective plane is a projective line, and any projective line satisfying enough axioms can be embedded as a line in some projective plane. This would be analogous to how any plane in a projective 3-space is a Desarguesian projective plane, while any Desarguesian projective plane can be embedded in a projective 3-space.

I have an idea for how one might do this, by axiomatizing the "quadrangular hexad" relation on a line in a projective plane; but before I try very hard, I'm looking for references where something like it has been tried before.

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    $\begingroup$ Something related to this might have been done in the theory of Moufang sets. (These are, more generally, related to algebraic groups of rank one, but in particular the projective line with its $\operatorname{PGL}_2$-action is an example of a Moufang set.) Could be that some axioms exist that single out the projective lines among the wild world of Moufang sets... $\endgroup$ – Matthias Wendt Oct 16 '15 at 6:41
  • $\begingroup$ Given 3 distinct and 4 distinct points in $\mathbb P^1$, we can use the first three to decide where $0$, $1$, $\infty$ are, and the latter four to specify a cross-ratio. That number should then be an 8th point. I have no idea what relations this 7-ary (partially defined) operation should satisfy. $\endgroup$ – Allen Knutson Oct 17 '15 at 3:18
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    $\begingroup$ @AllenKnutson That's similar to what I was thinking of. Note that once you've specified 0, 1, and $\infty$, you don't need to muck around with cross-ratios any more to get something projective; every point is already a single number, so you can just add and multiply them directly. This gives you some partial 5-ary operations, which can both be encoded using the functionality of the 6-ary "quadrangular hexad" relation. But at this point I'm mainly wondering whether anything like this has been done before; it seems a natural thing to try. $\endgroup$ – Mike Shulman Oct 18 '15 at 4:15
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Building on previous work by Paul Libois, and related to work by Libois' student Jean van Buggenhaut from 1969, Francis Buekenhout considered and solved this question in "Foundations of one Dimensional Projective Geometry based on Perspectivities" Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 43 (1975) 21-29. Note that his approach also adapts to lines in Moufang planes. See also F. Buekenhout and A. Cohen Diagram geometry Springer 2013, Section 6.2 Projective lines.

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    $\begingroup$ The book by Buekenhout and Cohen is much more easy to obtain. I just had a look at the copy in my shelf; section 6.2 quite cleary states that it is about classifying projective lines in thick projective spaces, which are Desarguesian. $\endgroup$ – Max Horn Nov 18 '15 at 7:37
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    $\begingroup$ @MikeShulman A line in a projective plane is a projective line in this sense if the plane is a translation plane with respect to this line. Most lines in non-Desarguesian projective planes are not projective lines in this sense. $\endgroup$ – Tim Penttila Nov 20 '15 at 14:22
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    $\begingroup$ (continued) This is for the restricted definition where there's only the hypothesis when the two points are equal. Buekenhout has replaced Desargues (little Desargues in the restricted sense) by the symmetry it induces on a line, and a general line in a non-Desarguesian plane won't have that symmetry. $\endgroup$ – Tim Penttila Nov 20 '15 at 14:32
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    $\begingroup$ The paper has five citations:Hirschfeld's book, the Buekenhout-Cohen book, J.Tits, Twin buildings and groups of Kac-Moody type 1992,PM Johnson Semiquadratic sets...1999 H Van Maldeghem Moufang lines 2007. None of them really take the idea much further. The last one is essntially studying the translation line in the Luneburg plane from this perspective. $\endgroup$ – Tim Penttila Nov 20 '15 at 14:46
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    $\begingroup$ I finally got a copy of the Buekenhout-Cohen book, which finally contains an actual definition of what he means by "perspectivity", which is not what I thought: he means the restriction to the line in question of a central collineation of the ambient projective space (an automorphism that fixes some hyperplane pointwise and all lines through some point setwise). There's still an exercise I have to do to convince myself that this works, but at least it seems more plausible. $\endgroup$ – Mike Shulman Dec 3 '15 at 20:20
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Following up on Matthias Wendt's comment, the language of Moufang sets is indeed a suitable axiomatic approach to (generalizations of) projective lines.

Formally speaking, a Moufang set is a set $X$ together with a collection of groups $U_x \leq \operatorname{Sym}(X)$ (one group for each $x \in X$), such that:

  • each $U_x$ fixes $x$ and acts sharply transitively on $X \setminus \{ x \}$;
  • the group $G := \langle U_y \mid y \in X \rangle$ permutes the $U_x$'s by conjugation.

The groups $U_x$ are called the root groups of the Moufang set, and the group $G$ is called the little projective group.

It is not hard to see that $X = \mathbb{P}^1(k)$ and $G = \mathrm{PSL}_2(k)$ has the structure of a Moufang set, but there are many more interesting examples, most notably those arising from semisimple linear algebraic groups of $k$-rank one.

It is possible to single out the genuine projective lines over fields among the Moufang sets. For fiels $k$ with $\operatorname{char}(k) \neq 2$, this was done by Richard Weiss and me (see section 6 of our paper "Moufang sets and Jordan division algebras", Math. Ann. 335 (2006), no. 2, 415–433); this result has been generalized to arbitrary fields (including the case $\operatorname{char}(k) = 2$) by Matthias Grüninger, where the result is somewhat more subtle ("Special Moufang sets with abelian Hua subgroup", J. Algebra 323 (2010), no. 6, 1797–1801).

At the risk of giving too much advertisement for my own papers, I can recommend the course notes "A course on Moufang sets", Innov. Incidence Geom. 9 (2009), 79–122 that I wrote together with Yoav Segev, for an introduction to the subject.


Edit: As requested in the comments below, I am adding some more details, in particular about the example with $X = \mathbb{P}^1(k)$ and $G = \mathrm{PSL}_2(k)$.

First of all, notice that the little projective group $G$ is generated by any two of the root groups, $G = \langle U_x, U_y \rangle$ for all $x,y \in X$ with $x \neq y$. So to give an explicit description of an example, it suffices to describe two of these root groups; all others are then obtained by conjugation inside $G$.

We now take $X = \mathbb{P}^1(k) = k \cup \{ \infty \}$, acted upon by $G=PSL_2(k)$, the elements of which I will denote with matrices with square brackets (determined up to a non-zero scalar), so $$ \begin{bmatrix} a & b \\ c & d \end{bmatrix} .x = \frac{ax+b}{cx+d} \quad \text{for all } x \in X.$$ Notice that $\operatorname{Stab}_G(\infty) = \left\{ \begin{bmatrix} a & b \\ 0 & a^{-1} \end{bmatrix}\right\}$ and $\operatorname{Stab}_G(0) = \left\{ \begin{bmatrix} a & 0 \\ c & a^{-1} \end{bmatrix}\right\}$. We now define the root groups $U_\infty$ and $U_0$ to be the group of unipotent elements of these point stabilizers, i.e. $$ U_\infty =\left\{ \begin{bmatrix} 1 & b \\ 0 & 1 \end{bmatrix} \mid b \in k \right\}, \quad U_\infty =\left\{ \begin{bmatrix} 1 & 0 \\ c & 1 \end{bmatrix} \mid c \in k \right\} .$$ The point is that it is now possible to forget about the matrix representation and even about the original ambient group $G$ all together, and only retain the corresponding permutations of $X$. We then get $$ U_\infty = \{ x \mapsto x + b \mid b \in k \}, \quad U_0 = \{ x \mapsto (x^{-1}+c)^{-1} \mid c \in k \} . $$ It is now not so hard to imagine that this description makes sense for more general algebraic structures than commutative fields only. And indeed, this works equally well for skew fields, octonion division algebras, and even more generally for Jordan division algebras. To make examples with non-abelian root groups, similar ideas make sense by replacing the multiplicative inverse by more complicated maps that "behave like a multiplicative inverse".

Another relevant comment, related to your first "ideal property": in the case of skew fields, for instance, it is only possible to recover the skew field up to opposition, i.e., in the case of $\mathrm{PSL}_2(D)$, we can recover the pair $(D, D^{\mathrm{op}})$ from the Moufang set, but not $D$ itself. (Here, $D^{\mathrm{op}}$ is the skew field with same underlying additive group as $D$, and with multiplication given by $x*y := yx$.)

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    $\begingroup$ Well, I also recommend those course notes, and I am not a co-author, so I think I am allowed to do that ;-) $\endgroup$ – Max Horn Nov 18 '15 at 11:37
  • $\begingroup$ Of course Moufang sets are not that helpful if one also wants to capture projective lines contained in projective planes with small or even trivial automorphism groups. But catching those sounds like a very difficult problem to me, given that it is not known if there are projective planes of non-prime-power order... So if one could characterize projective lines intrinsically via axioms, deciding whether the finite ones can have non-prime-power order ought to be a very difficult problem. $\endgroup$ – Max Horn Nov 18 '15 at 11:41
  • $\begingroup$ I don't suppose you could add a short description of what the groups $U_x$ are in the case of $\mathbb{P}^1(k)$, and ideally a synthetic description if $X$ is a line in a synthetic projective plane? This looks a lot like the definition in Buekenhout's paper, where I think the groups are supposed to be "central collineations", except that Buekenhout slices up the collineations according to their axis as well as center, so I'm wondering if the structures are the same. Does this work for projective lines over division rings, or in non-Desarguesian planes? $\endgroup$ – Mike Shulman Nov 18 '15 at 16:12
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    $\begingroup$ I was just saying I think the answer above would be enriched by a concise description of the construction, so a reader doesn't have to make their way through 20 pages of notes to piece together the definition. And octonion division algebras are one particular nonDesarguesian situation, but is there a synthetic construction that works in general? $\endgroup$ – Mike Shulman Nov 18 '15 at 19:43
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    $\begingroup$ Yes, it is the Buekenhout definition, when the two points are equal (and no hypothesis with the points distinct). The Moufang set people tend not to cite people who worked on the topic before Tits, probably because they didn't use the terminology of Tits. (So, for instance, John Faulkner's early work is also not often cited.) $\endgroup$ – Tim Penttila Nov 20 '15 at 15:07
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One defining feature of $\mathbb P^1(k)$ is that it provides a sharply 3-transitive permutation representation for $\operatorname{PGL}_2(k)$. I believe that the abstraction of projective line to "sharply 3-transitive permutation group" is the most studied one.

The characterization of sharply 3-transitive groups as groups of projectivities over KT-fields came up in an answer to Jacob Lurie's question Action of PGL(2) on Projective Space. That answer mentions that every sharply 3-transitive group is the "group of projectivities" of a KT-field $F$, but note that the correspondence goes both ways, and one can construct $F$ out of the permutation representation.

Judging by how there is an equivalence of categories of near fields and of sharply 2-transitive groups, I wouldn't be surprised if one can say something similar for KT-fields and sharply 3-transitive groups. A reference for this is

"Kerby, W., Wefelscheid, H. "Uber eine scharf 3-fach transitiven Gruppen zugeordnete algebraische Struktur." Abh. Math. Sem. Univ. Hamburg 37 (1972), 225–235.

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You might benefit from reading Section 5.3 of John Faulkner's book The role of nonassociative algebra in projective geometry AMS 2014. The results there may have been what you had in mind by 'axiomatizing the "quadrangular hexad" relation'. These are (almost) all very old theorems, appearing in (and mostly predating) Pickert's Projektive Ebenen book from 1955, one going back to Veblen and Young from 1910 and another to von Staudt from 1860. The basic underlying idea of using quadrangular sets goes back to Desargues in 1639. (It was Desargues who introduced the term involution into mathematics, in this context, often referred to as an involution of six points.) But Desargues' work is an elaboration and extension of results of Pappus from c.340.

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  • $\begingroup$ Yes, Faulkner's lemma 5.6 and the first sentence of his lemma 5.7 are some of the axioms I would include. I didn't know that uniqueness of the sixth point is equivalent to Desargues (though it doesn't surprise me). But of course his section 5.3 is not itself what I was thinking of doing, since he is working in an ambient projective plane throughout, rather than axiomatizing a hexary relation on a given line. $\endgroup$ – Mike Shulman Dec 3 '15 at 22:36

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