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Modelling Finite Geometries on Surfaces

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Modelling Finite Geometries on Surfaces View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Discrete Mathematics 244 (2002) 479–493 www.elsevier.com/locate/disc Modelling ÿnite geometries on surfaces Arthur T. White Western Michigan University, Kalamazoo, Michigan 49008, USA In Honor ofProfessorToma zÄ Pisanski Abstract The usual model ofthe Fano plane has several deÿciencies, all ofwhich are remedied by imbedding the complete graph K7 on the torus. This idea is generalized, in two directions: (1) Let q be a prime power. E3cient topological models are described for each PG(2;q), and a study is begun for AG(2;q). (2) A 3-conÿguration is a geometry satisfying: (i) each line is on exactly 3 points; (ii) each point is on exactly r lines, where r is a ÿxed positive integer; (iii) two points are on at most one line. Surface models are found for 3-conÿgurations of low order, including those ofPappus and Desargues. Special attention is paid to AG(2,3). A study is begun ofpartial geometries which are also 3-conÿgurations, and fourgeneral constructions are given. c 2002 Elsevier Science B.V. All rights reserved. Keywords: Finite geometry; Projective plane; A3ne plane; Partial geometry; Conÿguration; Surface; Voltage graph 1. Introduction A ÿnite geometry can be modelled by specifying its point set P and then giving the line set L by listing each lines as a subset of P. This abstract model reinforces the interpretation ofthe geometry as a block design, or as a hypergraph. But it lacks the geometric =avor available through the use of0-dimensional subspaces ofsome Euclidean n-space to model P, and higher dimensional subspaces to model L. The usual model ofthe Fano plane, which has achieved logo status in combinatorics and is shown here in Fig. 1, is an attempt to provide such a geometric rendering. But this model, despite its popularity, has serious defects: (1) The line {1; 2; 4} is rendered di>erently, yet there is nothing in the axioms for this geometry to distinguish one line from the others. In fact, Z7 acts regularly on L, as the labelling we have used reveals. (2) Each ofthe other lines appears to have a midpoint, but there is no concept of ‘betweenness’ in the axioms. E-mail address: [email protected] (A.T. White). 0012-365X/02/$ - see front matter c 2002 Elsevier Science B.V. All rights reserved. PII: S0012-365X(01)00069-3 480 A.T. White / Discrete Mathematics 244 (2002) 479–493 Fig. 1. (3) One gets the impression, by looking at small neighborhoods ofthe points, that three points are on three lines each, three are on ÿve lines each, and the ‘central’ point (another inappropriate feature) is on six lines. (4) There are three extraneous intersections ofedges in the graph-like ÿgure. These have no meaning in the geometry, and they give the greatest o>ense to a topological graph theorist. The Fano plane model ofFig. 2 remedies all ofthese deÿciencies, by imbedding the complete graph K7 in the torus. The seven vertices depict the points ofthe Fano plane, and the seven unshaded regions represent the lines—by means ofthe points appearing in the respective boundaries. The seven shaded regions ofthe K7 imbedding are hyperregions; they are what remain in the surface when the image of the geometry is removed. The K7 imbedding is shown as a 7-fold covering space over a one-vertex, three-loop voltage graph, also imbedded in a torus. All the information about the former structure is contained in the latter, as can be discerned from the ÿgure. Thus many imbeddings ofinterest are constructed by liftingan appropriate voltage graph imbedding, reversing the arrow ofFig. 2. See [5], or Chapter 10 of[8], fordetails of voltage graph constructions. We claim that the surface model of the Fano plane just given is the most e3cient such model possible, that the Fano plane has genus one. Formally, this is established by means ofthe Levi graph (P ∪ L as the vertex set, edges given by incidence). The Levi graph ofthe Fano plane has girth six (it is the Heawood graph), and no graph ofgirth six and minimum degree at least three is planar (a plane graph with such a minimum degree must have a region with ÿve or fewer sides). An easy modiÿcation of Fig. 2 (add a new vertex in interior ofeach unshaded region above, join it to each boundary vertex ofthat region, delete all the K7 edges) provides the Levi graph ofthe Fano plane, in the torus, completing the calculation ofgenus one forthis geometry; see [9]. (Conversely, an imbedding ofa Levi graph can be modiÿed to obtain an imbedding of the geometry on the same surface. Thus a geometry and its Levi graph have the same genus.) Informally, the model displayed cannot be improved upon, as all hyperregions are triangular. A.T. White / Discrete Mathematics 244 (2002) 479–493 481 Fig. 2. In this paper we generalize the Fano plane, ÿrst to the class ofÿnite projec- tive planes PG(2;q) and the related a3ne planes AG(2;q), and then to the class of 3-conÿgurations. All geometries considered have the property that two points belong to at most one line. In each case we seek an e3cient topological model into a surface, a pseudosurface, or a generalized pseudosurface. Examples should make the distinction clear, but see [8] for precise deÿnitions and other background material. 2. Finite projective planes The Fano plane is the projective plane ofsmallest order; it is PG(2 ;q), for q =2. These Desarguesian planes exist ifand only ifthe order q is a prime power. In [9] topological models were constructed for all these geometries, as summarized below. 482 A.T. White / Discrete Mathematics 244 (2002) 479–493 The model for PG(2;q) has the feature that Zq2+q+1, which acts regularly on both the point (vertex) set and the line (region) set, also acts regularly on each orbit ofthe set ofhyperregions. Moreover, the models foundare optimally e3cient (in terms of minimizing genus, or maximizing characteristic), for that feature. The graphs used are Cayley graphs G ( )for = Zq2+q+1 and consisting of q + 1 generators which lead to a perfect di>erence set. (1) For q ≡ 2 (mod 3); PG(2;q) is modelled on an orientable surface of genus 1 + (q − 2)(q2 + q +1)=3, with all hyperregions triangular. (2) For q ≡ 1 (mod 3); PG(2;q) is modelled on an orientable surface of genus 1 + (q − 1)(q2 + q +1)=3, with q2 + q + 1 hyperregions pentagonal and all others triangular. (3) For q ≡ 0 (mod 3); PG(2;q) is modelled on an orientable pseudosurface of char- acteristic (3 − 2q)(q2 + q +1)=3, with q2 + q + 1 hyperregions quadrilateral and all others triangular. Figueroa-Centeno [2] greatly extended these ideas, modelling PG(m; q)form+1 prime, and q a prime power. 3. Finite ane planes A ÿnite projective plane oforder n is an (n2 + n +1;n2 + n +1;n+1;n+1; 1)-BIBD. (We give, in sequence, the number v ofobjects (points), the number b ofblocks (lines), the replication number r, the block size k, and the value: every two objects are in exactly common blocks.) By removing one line and all points on that line, we obtain an (n2;n2 + n; n +1;n;1)-BIBD, which is an a3ne plane. Ifwe start with n = q (a prime power) and PG(2;q)—only prime power orders are known to give a projective plane, but for some of these PG(2;q) is not the only projective plane of that order—we get the a3ne plane AG(2;q). It is these geometries we seek to model next. The construction ofAG(2 ;q) by deletions from PG(2;q) indicates that there is a deletion model for each AG(2;q), readily obtainable from the PG(2;q) models ofthe preceding section; see [9]. In fact, for q ≡ 0 (mod 3)—that is, for 3 the base prime— the pseudosurface model can be simpliÿed slightly, by removing q + 1 ofthe singular points ofthe topological space, corresponding to the points ofthe deleted line in the geometry. But none ofthe deletion models is satisfactory.Not only does the imbedded graph have only the trivial automorphism; now the lines are no longer depicted by closed walks bounding regions, but by paths partially bounding regions. Moreover, the euler characteristic is too low. Thus we seek an alternative to the deletion model, for AG(2;q). For q=2, deletions from Fig. 2—say, the line {0; 1; 3} and those three points—leave AG(2,2) on the torus. But this geometry is planar, as it is readily modelled by K4 (on points 2; 4; 5; and 6). For q = p, an odd prime, the spherical voltage graph ofFig. 3, A.T. White / Discrete Mathematics 244 (2002) 479–493 483 Fig. 3. using generating set = {(0; 1); (1; 0); (1; 1);:::;(1; p − 1)} for group = Zp × Zp, lifts to an imbedding ofthe Cayley graph G ( )onSk , the orientable surface of genus k =1+p(p−2)(p+1)=2. The p regions covering the loop carrying generator (0,1) are the vertical lines in AG(2;p), coordinatized. The p regions covering the loop carrying (1;m) are the lines ofslope m, for 0 6 m 6 p − 1. This gives the well-known resolution ofthe lines ofAG(2 ;p) into p + 1 parallel classes, of p lines each, each class partitioning the point set.
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