Chain Geometry Determined by the Affine Group

Chain geometry associated with an affine group and with a linear group is studied. In particular, closely related to the respective chain geometries affine partial linear spaces and generalizations of sliced spaces are defined. The automorphisms of thus obtained structures are determined.


Introduction
The most general attempt to chain geometry associated with a family of transformations was presented in [7] (see also [4]). Following this approach we consider incidence structures whose blocks are the graphs of a family of bijections. Restricting the class of admissible families of bijections we arrive to some more regular and rich geometries. Particularly, 2-rigid and 3-rigid families are commonly considered (cf. e.g. [1,6,10]) and the corresponding incidence geometries are well known.
Still, there are quite well known transformation groups which are neither 2-nor 3-rigid and whose graphs yield interesting incidence structures. In this note we discuss geometry associated with the group of linear bijections of a vector space and the group of affine transformations of an affine space. It turns out that in terms of the corresponding incidence structures with graphs as blocks one can define other incidence structures with lines, which can be obtained from affine and projective spaces by omitting some points and lines.
Such an incidence structure is called a reduct (of the underlying affine/projective space).
The first arising class consists of line-reducts of affine spaces obtained by deleting the lines parallel to one of a pair of fixed complementary affine subspaces. Formally, one can consider the construction of such reducts as a generalization of the construction presented in [2]; at any rate corresponding reducts are partial affine spaces (cf. [3,9]). The second class consists of structures obtained by deleting from a projective space the points on a pair of complementary subspaces and the lines which cross any of these subspaces. Again, this construction generalizes the known construction of a sliced space (cf. [5,8]). The underlying affine and projective space can be recovered from the corresponding reduct. From this we easily characterize the automorphisms of the structure of graphs of the linear and of the affine bijections.

Definitions
Let us start with the notation and notions used in the paper. Let W be a vector space; we write Each f ∈ ΓA(W) is associated with a unique automorphism f of the underlying division ring such that f = τ ϕ, τ is a translation, and ϕ ∈ ΓL(W) is f -semilinear. Let V = (V, θ, +, ·) be a fixed vector space over a division ring F and Y = V × V. In a standard way with a family of transformations F ⊂ V V ⊂ ℘ (V × V ) we associate the structure M(F) of the graphs of F: In accordance with general theory (cf. e.g. [1, Chap. III, §4]) the following holds In the context of general chain geometries we write a + a when a , a are on a block in L + , and a − a when they are on a block in L − .
For points a, b of M(F) we say that they are joinable and we write a ∼ b In what follows we shall be mainly concerned with the structure M(F), where F = GA((V)) and F = GL((V)). Actually, we shall investigate the structure which is a bit weaker (formally) than M(F). Let F = GA(V); with a = (u 1 , u 2 ), b = (v 1 , v 2 ) and u i , v i ∈ V we have a ∼ b iff there is an affine map f such that f (u 1 ) = u 2 and f (v 1 ) = v 2 . Since the group GA(V) acts 2-transitively on V we get

Chains Determined by the Affine Group
In view of Lemma 3.1, the following conditions are, clearly, equivalent: This gives immediately that for a = b, a ∼ b we have In particular, we have

Two-Directions Reduct of an Affine Space
Let a vector space Y be the direct sum of its two subspaces V + and V − , Without loss of generality we can write Y in the form In mathematical practice, these two approaches are frequently mixed and the spaces in pairs Write L 0 for the class of lines of the affine space AG(Y) =: A that are not parallel neither to V + nor to V − and set Then Corollary 3.2, (2), and Lemma 3.3 state the following Clearly, B(V + , V − ) is a partial linear space. For points a, b of an arbitrary partial linear space D we write a ∼ b when they are collinear i.e. when they are on a line of D. In this subsection we shall apply this definition to the points of B(V + , V − ) =: B. In the particular case when B is determined by the structure M * (GA(V)) as in Corollary 3.4, the binary collinearity of B and the relation of joinability in M * (GA(V)) introduced in Sect. 2 coincide (cf. Lemma 3.1), so we can use the same symbol to denote them. What is more, Lemma 3.1, Corollary 3.2, and formula (1) remain valid in an arbitrary two-directions reduct B(V + , V − ).
Let P be the class of affine planes in A. The following is just an easy observation.
Lemma 3.5. Let π ∈ P and let B be the set of affine lines that lie on π and do not belong to L 0 . Then one of the following holds: is the union of two parallel pencils on π, (iv) all the lines on π are in B.
In each case except (iv), in which π V + or π V − , π contains a triangle with the sides in L 0 . Assume that dim(Y) ≥ 3. Then for every affine line L / ∈ L 0 there are planes π 1 , π 2 such that L = π 1 ∩ π 2 and π i satisfies either (i) or (ii) for both i = 1 and i = 2. Lemma 3.6. Let π ∈ P and let L 1 , L 2 , L 3 ∈ L 0 yield a triangle in π. Assume that the coordinate division ring of Y contains at least 5 elements. Then ⎛ Proof. Set X = L∈L0 : |L∩(L1∪L2∪L3)|≥2 L ; clearly, X ⊂ π. Let a 1 , a 2 , a 3 be the vertices of our triangle, a i / ∈ L i for i = 1, 2, 3. Let x ∈ π. By Lemma 3.5, there are at most two affine lines through x not in L 0 and thus there is at least one line M in L 0 that is contained in π, goes through x, and crosses L 1 in a point c distinct from a 2 , a 3 . Clearly, c / ∈ L 2 , L 3 . Then M ∦ L 2 or M ∦ L 3 , so M crosses a second side of our triangle in a point distinct from c and thus x ∈ X .
Since Lemma 3.6 plays an important role in further investigations, from now on till the end of Sect. 3 we assume that the coordinate division ring of Y is not GF (2), GF (3), and GF (4).
In view of Proposition 3.7, a characterization of the whole group Aut(B(V + , V − )) is easy.
for all x 1 , x 2 ∈ V .
Finally, note that an automorphism F defined by (5) can be written in

Chains Determined by the Linear Group
As an immediate consequence of Lemma 3.3 we obtain the following i.e. it is the one-dimensional subspace of Y spanned by a. Each one dimensional subspace of Y not contained in V + ∪ V − can be presented in this way.   Proof. Assume that a ∼ b. In view of Lemma 4.2 this means that there is a scalar λ such that λu 1 = v 1 and λu 2 = v 2 (or λu 1 = v 1 and λu 2 = v 2 ). Consider the first case. Let β = 0 be arbitrary, α = −βλ, and c = αa + βb. Then U c = (θ, β(v 2 − λu 2 )) ∈ V + and c = (θ, θ). In the second case, analogously, we find a non zero c ∈ U ∩ V − . Thus (i) implies (ii). Now, let c ∈ U ∩ V + be non zero i.e. c = (θ, v) = αa + βb for some v = θ and scalars α, β. In particular, θ = αu 1 + βv 1 and v = αu 2 + βv 2 . Either α = 0 or β = 0, as v = θ. Assume that β = 0, then v 1 = λu 1 for a scalar λ. If there were v 2 = λu 2 we get a = λb and either λ = 0 (and, consequently, a = θ) or a = b , which both contradict the assumptions. So, v 2 = λu 2 and, by Lemma 4.2, a ∼ b. In the case α = 0, analogously, we end up with a ∼ b. Finally, analogous reasoning proves (i) when there is a non-zero vector c ∈ U ∩ V − .

Finally, as in Lemma 3.3 one can compute
only can be presented in this form.

Two-Holes Sliced Space
As in Sect. 3.1 we fix a vector space Y being a direct sum of two its subspaces V + and V − . And, as in Sect. 3.1 we do not assume that dim(V + ) = dim(V − ). Let θ be the zero vector of Y. Consider the projective space P = P G(Y) and its two subspaces Then  The next step is to show that P can be reinterpreted in terms of its reduct T. This needs some work.
Let us call the projective points in S proper and those in H + ∪ H − improper. Let L be a line of P not contained in H + ∪ H − . We write L ∞ + for the point in L ∩ H + and L ∞ − for L ∩ H − , if nonempty; otherwise L ∞ + (L ∞ − resp.) is ∅ simply. Let us write L ∈ L i when |L \ S| = i. Consequently, L ∈ L 0 when L ∞ + ∪ L ∞ − = ∅, L ∈ L 1 when either L ∞ + = ∅ and L ∞ − is a point or L ∞ − = ∅ and L ∞ + is a point, and L ∈ L 2 when L ∞ + and L ∞ − both are points.
Let A be a plane of P not contained in H + ∪ H − . Analogously we use the symbols A ∞ + and A ∞ − (note that following such an approach we frequently identify a point with the set consisting of this point. We believe that this should not lead to a misunderstanding, though). Note an evident Write Π i for the class of planes of P on which i points are outside S with i = 0, 1, 2, and let Π = is a point, and A ∈ Π 2 when A ∞ + and A ∞ − both are points. Let us warn that Π is not the set of all the planes of P not contained in H + ∪ H − .
With elementary geometrical reasoning we obtain two subsequent lemmas.
Lemma 4.7. Let A be a plane of P. Assume that the size of the lines in P is at least 4.

(i) If A ∞ + is a line or A ∞ − is a line then A does not contain any line in T .
(ii) Assume that neither A ∞ + nor A ∞ − is a line. Then A contains a triangle of T (i.e. with the sides in T and the vertices in S).
(iii) Let A contain a triangle of T with the sides L 1 , In view of Lemma 4.7, as in Sect. 3.1 from now on till the end of Sect. 4 we adopt the assumption ( ).
Let ∼ denote the binary collinearity relation of T and ∼ be its complement.

Lemma 4.8. Let
A be a plane of P.
Then the noncollinearity relation ∼ is transitive on A∩S and thus the relation ∼ ∪ = is an equivalence relation. Its equivalence classes are exactly the sets L \ Q, where L is a line on A not in T . Each line L of P contained in A and not in T is in L 1 .
(ii) Let A, Q be as in (i) and (iv) Let A, Q be as in (iii) and From Lemma 4.7, A ∈ Π when A contains a triangle in T, and the class Π is definable in T or, to be more precise, the class To complete our course we need the following (proved with an easy though tedious linear combinatorics) Then either the projective plane A through L 1 , L 2 is in Π or there is a line L 3 ∈ L 1 through q, L 3 = L 1 , L 2 such that both planes: through L 3 , L 1 and through L 3 , L 2 are in Π.
The following analogue of a parallelism M 1 M 2 is definable in terms of geometry of T for M 1 , M 2 in the class M 1 The relation is an equivalence relation.
For M ∈ M denote by M the line of P which contains M and write Thus the equivalence classes of the relation ⊂ M 1 × M 2 correspond to the points in H + ∪ H − , which were deleted from P when T was defined. Thus, we have re-defined the point set of P in terms of T. We need to re-define the omitted lines of P: those, which are not entirely contained in H + ∪ H − correspond to the elements of M ∪ T . So we need to re-define the incidence of the improper points and the elements of M. By the above, it is immediate for the elements of M 1 .
Next, an analogous parallelism (say, ) contained in M 1 × M 2 can be defined with the property: 'M 1 M 2 iff M ∞ 1 ⊂ M ∞ 2 '. Completing the lines in M by their improper points we obtain the class T ∪ L 1 ∪ L 2 of all the lines of P not entirely contained in H + ∪ H − . Finally, it is a trivial trick to re-define the lines on H + ∪ H − as the sets of the improper points of the lines on suitable planes spanned by the lines in L 1 . Thus, finally, we arrive to the following
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