Dimensional orbits in flat projective planes

Math. Z. 122, 117-124 (1971) 9 by Springer-Verlag 1971

1-Dimensional Orbits in Flat Projective Planes* HANSJOACHIM GROH

1. Introduction In [8, Lemma 10, 11], Salzmann showed that the 1-dimensional orbit of the automorphism group a P of a "hyperbolic" projective plane P is an oval by using the fact that a P ~ PSL 2 (R). The purpose of this paper is to investigate for arbitrary fiat projective planes P orbits of 1-dimensional subgroups A of aP. Our main result (Theorem 3) is that such orbits, unless they are contained in a line, are always ovals if A ~ S1 (Circle Group), including the above result. If A mR (reals), the situation is more complicated, but the orbits are either "spirals" or "nearly" ovals (compare also [1, 2] for the Desarguesian case). These results will be an essential tool in the classification of fiat projective planes with dim a P = 2.

2. Topological Properties of 1-Dimensional Orbits Throughout this paper, let (P, L) be a fiat projective plane, i.e. a topological projective plane ([5]) whose point space P is a 2-manifold. The only 2-manifold which is possible is the space P2 of the classical projective plane, and the lines must be Jordan curves, i.e. homeomorphic images of $1 ([-6]). For simplicity, we will often denote the plane by its point space P. The group: a P of all automorphisms, given the compact open topology, is a locally euclidean group of dimension < 8 and acts as topological transformation group on P and L ([,7, Satz 4.1 and Satz 3.1]).

Definition. An orbit of a subgroup A of a P is called trivial if it is contained in a line. Notations. For p~P, the set of lines containing p is denoted by p,. "-~" means isomorphic (e.g. for topological groups), but "homeomorphic" is abbreviated by ~. We write G < H iff G is a subgroup of H. If I is a subspace of X ~ R and a, b~l, ]a, b [~ stands for the intersection of the open interval { x ~ X ] a < x < b } with I. Similarly [-a, b[,l= {a} u ] a, b I-z, etc.

* Research supported by the National Research Councilof Canada, Grant A7336.

118

H.Groh:

L e m m a l I. If a P > A ~ S 1 o1" R, then A fixes a point p and a line I. For A ~-$1, p and l are uniquely determined, pq~l, and A is transitive on I. For A g R , precisely one of these cases holds: (1) p and I are uniquely determined, p$1. The group Z of(p, l)-dilatations in d is infinite cyclic, d is transitive on I. (2) Every fixed point of A lies on a fixed line.

Proof For the first statement, let flEA be an element of infinite order. By the divisibility of A, we can find recursively a sequence {fn} of elements satisfying f~+l=fn. All f, have again infinite order, and in particular the subgroup G generated by them is not cyclic and therefore dense in A. The set F(fn+ 1) of fixed points of f , + l is contained in F(f,). Also F(f,)4= ~, as every homeomorphism of P2 has a fixed point. Therefore, as the F(f,) are closed and have the finite intersection property, the compactness of P implies c~ F(f,) 4: 0, which implies a universal fixed point for G and therefore for G = A. If A -_ S 1, p a fixed point, I a fixed line, then the classification of all transitive, locally compact, connected, effective transformation groups on 1-manifolds by Brouwer ([3]; see [9, Hilfssatz 1.9] for modern terminology and proof), in the following quoted as Brouwer's theorem, implies that A is transitive on p . ~ S t . Therefore p~l, A is transitive on l, and l is unique. If A -~R, assume that there is a point p fixed by A which lies on no fixed line. Then A must be transitive on p~ and as above, there is only one fixed line I of A, and A is transitive on 1. By Brouwer's theorem, A cannot be effective on po ,~ $1. Its ineffectivity subgroup Z, by definition the group of all (p, /)-dilatations in A, is therefore a nontrivial closed subgroup of A, and hence infinite cyclic. Lemma 2. If A
Proof We first show that A is strictly transitive on M, i.e. for each q s M , the evaluation map A ~6 ~-~q~EM is injective. Let p be a point fixed by A, and let (~Aq, the stabilizer of q. If d ~S~, then because of Brouwer's theorem, A fixes not only the line p v q but belongs to the subgroup of A ineffective on p~ Thus ~ is a dilatation which fixes q. As M is nontrivial, qr {p} u l, where l is the line fixed by A, and therefore 6 = 1. If A-~ R, the same consideration applies for case (1) of Lemma 1. In case (2), as M is nontrivial, (p v q)a must be a 1-dimensional subspace of p~ As A fixes a line through p, this subspace is homeomorphic to R. By Brouwer's theorem, d must be strictly transitive on it, implying A q= 1. The proof of M ~ A is now immediate for A ~ S~ because of the compactness of $1. For A ~ R , we have in addition to show that q6~---~q implies 6,-~ 1, i.e. for each neighborhood U of the identity 1 of z~, q is not an accumulation point If P is not "hyperbolic" the existenceof a fixedpoint is a consequenceof [9, Satz 1.3], The statements for A~S~ resp. zl ~R, case (1) are known by [8, Lemma6] resp. 19, Hilfssatz1.6]. We give, in this context,a uniform proof for the differentcases.

1-Dimensional Orbits in Flat ProjectivePlanes

119

of q~,V. We may assume that U is connected and (p v q)e#p.. If q were an accumulation point of qa-.v, then p v q would be an accumulation point of (p v q)~" v. In case (2) of Lemma 1 however, Brouwer's theorem implies that (p v q)V is open in p, and disjoint to (p v q)a-.v, which contradicts the above assumption. For case (1) of Lemma 1, the same argument shows that q can be at most an accumulation point of qA where A = U { U z I z ~ Z \ { 1 } } . L e t / b e the line fixed by A, and let b be the connected component of (p v q)'-.({p} w I) which contains q. Let Z + be the subgroup of Z mapping b into itself. Z + is again an infinite cyclic group; let g be one of its generators. Since Z + consists of dilatations, each of its nontrivial elements is a fixed point free and therefore orientation preserving homeomorphism on b~R. F r o m this action it follows that the only point of qZ+ "between" qg-~ and qg, i.e. contained in the open g-I interval ] q , qg [b, is q. The endpoints of the closed interval U = Ec5~,~2-1 determine lines li =(p v q)ai with qr l~. For each zeZ, qOZis an arc ( ~ [0, 1]), which has precisely its endpoints in common with ll and Iz. Therefore qVz is contained in one of the components of P ~. (l w l1w 12).Each of these is, by the Schoenflies theorem (see e. g. [4, p. 175]), open and homeomorphic to R 2, and is, as a consequence of the Jordan curve theorem, decomposed, by each qVZ contained in it, into two open components. Let V be the component of P'-. (l w 11w 12) containing q. Then, as V ~ (p v q) = b, qW meets V i f f z ~ Z +. In V, qVZ is "between" qW, and qW~ (i.e. qW is contained in the component Vu of V ' , q w` containing qWj, i#:j) iff, in b, q~ is between q~' and q~. Putting q = g - ~ , z2=g, Va2~ Vii is therefore a neighborhood of qV disjoint to all qW with z # 1, which proves the Lemma. Corollary. Each element of A which is not an axial automorphism fixes precisely the same points and lines as A.

Proof By Lemma 2, only points q in trivial orbits might be fixed by 1 @f6A. For A ~$1, and in case (1) of Lemma l for A ~ R , ifq is not the fixed point of A, then it is on the line fixed by A. This implies that f i s axial. In case (2) of Lemma 1, let qa ~ I where I is a line fixed by A. As A fixes a point of l, qA is either {q} or ~ R. In the second case, Brouwer's theorem implies that f cannot fix a point of q~. Lemma 3. I f A ~aP, A _~R, and M is a nontrivial orbit, then every point x of M \ M is either fixed by A or lies on a line fixed by A.

Proof In case (1) of Lemma 1, let q e M be in the component b of p v x ' - . ({p} ~ l) which contains x. Here we use the notations of the proof of Lemma 2. From this proof it follows that x is contained in some open set Vu of V with Vuc~ U==r for all z~Z, and therefore Vuc~M=0, a contradiction. In case (2) of Lemma 1, let p be a fixed point of A. Then, if x:#p and ( x v p ) a + x v p , (x vp) ~ is a proper open interval ] l~, l2 [- ofpo. Let q~M and {%} = {qa.}, 6,eA, be a sequence converging to x. Because ofx(iM, {q,} can have no accumulation point in ] I1, 12 [, implying p v x e {l1, I2}.

120

H.Groh: 3. Geometrical Properties: Secants

Lemma 4. Let A ~ a P , A ~-$1 or R. Let M be a nontriviaI orbit and g line. Then intg(M n g)=0. Proof. If the interior relative to g of M n g is nonempty it contains a subset U,.~R. By Lemma 2, M is a 1-manifold, and therefore U would be open also in M, i.e. intM(Mc~g)+~). Let Pl,P2 be two different points of U. Since A is a topological transformation group on M, there exists a neighborhood W of 1 in A with {Pl, P2} w~ U. Hence gV=(p 1 v p2) U= {g} which implies ga= {g} as W generates A. But then M _ g, i.e. M is trivial. Theorem 1. a. Let P be a flat projective plane, A a subgroup of its automorphism group, and assume that either (a) A _~$1, or (b) A ~ R and every fixed point of A lies on a fixed line. Then if M is any nontrivial orbit of A, and g is any line, ]M c~ g] <2 holds. Proof. In case (a), let p (resp. l) be the unique (Lemma 1) point (resp. line) fixed by A. Then P ' = P \ l ~ R 2 and g ' = g c ~ P ' ~ R . Let Pl,P2 be two different points of g n M = g' ~ M. By Lemma 4, there exist points q~ + q2 (~ [Pl, P2]g' such that [ql, q2]g' n M = {ql, q2}. By the Jordan curve theorem M ~ S 1 decomposes P' into two components, and ] ql, q2 [g,, being disjoint to M, is contained in one of them, denoted by C. Furthermore, as a consequence of the "0-curve theorem" (Whyburn [10, III 1.4, 1.5], ] ql, q2 [-g' decomposes C into two connected components C~, C2, such that the boundary of C~ in P' consists of [ql, q2]g, and precisely one of the two arcs determined by {ql, q2} on M~S1, and the boundary of C2 consists of [ql, q2]g' and the other arc. As A is connected, and as P ' \ M has only two components, A fixes each of them. From the action of A on M it follows that there exists a neighborhood U of 1 in A such that for each 6e U \ {1}, the pair {qq, q~2} separates the pair {ql,q2} on M ~ S ~ . As ]q~,q~2[=]ql,q2['~C, this implies, in view of the preceding paragraph, that ]q~, q~[ meets both C~ and C2. Therefore ]ql,

q2 Eg,~ ] q~l,q~2[g, * ~. Assume now that J M ~ g l > 3 . Then, by the above procedure, we can find pairs {ql, q2}, {q], q~} of different points with ] q~, q2 [g, c~]qi, q~ I-g,=9, and neighborhoods U, U' of 1 in A such that for 6 ~ U ~ U " - , { 1 } we have ]ql, q2[_gc3]q~, q~2[g,+(~ and ]ql, q'2[g'~ ]q't ~, q'2~[@O. This implies, however, [gng~ and therefore g=g~. Hence g is fixed by U and therefore by A, which implies M __g, a contradiction. Case (b) can be proved in analogy to case (a), with a few modifications. We therefore mention only these. P' is now defined as follows: Let p be a point fixed by A. Then for qosM, { p v q l q e M } = { p v q ~ is a proper (by assumption of (b)) open interval of p. ~ $1. Let la, I2 be the boundary elements of it (l 1= I2 may occur), and let P' be the component ofP \ (ll ~ 12)which contains M. We then have P ' ~ R 2, and g ' = g ~ P ' ~ R . By Lemma2, M ~ R , and by Lemma 3, M is closed in P', so that Jordan curve theorem and "0-curve theorem" are applicable. The boundaries of Ca and C2 are now [ql, q2]M resp. the closure of the union of the two other components of M \ {q~, q2}.

1-Dimensional Orbits in Flat Projective Planes

121

Theorem 1. b. Let P be a flat projective plane, R ~- A <.a P, and assume that A fixes a point p such that no line through p is fixed. Let M be a nontrivial orbit, q6M. Then there exists a neighborhood U of q such that t(M c~ U)c~gi < 2 for each line g. In fact, U can be chosen so that M c~ U is an open interval. Proof There exists ql ~ M and a generator z of Z such that q ~ b = ] ql, qZ [M. Abbreviate qZ = q2, ql v q2 = m , and let 1 be the unique ( L e m m a 1) line fixed by A. Then b is contained in one of the two c o m p o n e n t s into which m divides

m

qff

M P " . 1. F o r otherwise b c~ m:~9, which implies the existence of an element 3~A with q~ = q' ~ b c~ m. Because of p v ql = P v q2 = P V q ' , this would imply c~E Z, c~between 1 and z in A ~ R, which is impossible as z is a generator of the infinite cyclic group Z . - W e n o w have [xl, x z ] g . . ~ m = 9 for each line g which has two different points, Xa,X2 in c o m m o n with b: Assume x e ] x ~ , x 2 [ g , zc~m. Then x 1 and x2 are in two different c o m p o n e n t s of (p'-. l)'-. {x}, which in turn are in different c o m p o n e n t s of ( P \ l ) \ m . H o w e v e r we just showed that b is contained in one of these, a contradiction to x~, Xzffb. We n o w a p p l y the same p r o o f as for T h e o r e m 1.a on P ' = P \ I, where the r61e of M in T h e o r e m 1.a is played here b y j = b ~ [q~, q2]m-. ~. F o r applicability, we need only show that for each line g and xl, x2egc~b with [xa, X2]g..l~b= {xa, x2}, there exists a n e i g h b o r h o o d V of the identity in A with sVc~j=9 where s = ] Xl, x2 [g.. 1. First, we show that V~= ] z -1, z [4 is a n e i g h b o r h o o d of the identity with sV'c~ b =9- We have bV'= {~w b~u b ~ '. As the action of A on po is equivalent to the action of S~ on S~, it follows that for each line n through p, z interchanges the c o m p o n e n t s of (n'-. 1)'-. {p}. Applying this to n = q v p, we have that q~ is in the c o m p o n e n t of (P \ l ) \ m not containing b. This c o m p o n e n t therefore contains all of b ~ (and, similarly, all of b~-'). Assume n o w x~s6c~b for some 5 ~ V 1. Then x ~ ~ s c ~ b ~-2. As s c a b = 9 , we have sc~(b v'\b)+-9. H o w e v e r then s meets b \ b___m, or b : u b ~ ~ and therefore both c o m p o n e n t s of (P'-. l)'-. m. Hence s meets m in either case, a contradiction to the first paragraph.

122

H. Groh:

Secondly, the following is easy to see: Given a topological transformation group G on X, an open subspace W of X and a c o m p a c t set C__ W, there exists a n e i g h b o r h o o d V2 of the identity with C v2 c_ W. Applying this to G = A, X = P, W = the c o m p o n e n t of ( P \ l ) \ m containing s, C = g , we obtain a V2 with s V2n m = 0. Hence for V= 1/1 n V2 we have s v c~j = O. T o find a U, define h = ] q l , q~2[m\~. Then U = h 11'~ is open. F r o m M n h = {ql} it follows that M c~ U = q111'~ = b. 2--2

L e m m a 5. Let A
Proof In case (1), by L e m m a 3, M c _ M u { p } v o l , so that only {p}vol~_M remains to be shown. F o r x e l, the line g = p v x contains a point q of M. Then it is easy to see (for example from p r o o f of L e m m a 2 for case (1) of L e m m a 1) that qZc__g has p and x as accumulation points. In case (2), M \ M = C \ C where C is the c o m p l e m e n t of any c o m p a c t subset K of M. Choosing K to be a closed interval of M, we have from M ~ R that C = M \ K is the union of two connected c o m p o n e n t s C1, C 2. Assume Xl, x2~CI'~. C 1 with x l @ x 2. Let g be any line neither containing x 1 nor x2. F o r case (2), T h e o r e m 1 implies ]g n M] < 2, so that there exists a closed interval K ' ~ _ K u ( g c ~ M ) of M. D e n o t e the c o m p o n e n t s of M \ K ' by C], C~, sttch that C'~n Cg@O. Then C i " - C i = C ' i \ C'i. Let 1 be a line whose trace in P \ g ~ R 2 separates x 1 and x 2. Then from C'I~_P\g , the connectedness of C~, and x~, x 2 e C[ \ C a it follows that [C] c~ l[ > N o , a contradiction to T h e o r e m 1. Hence I C i \ Ci[ < 1. --

!

4. Tangents, Ovals, Spirals Definition. Let P be a projective plane, A _ P. A line t is called a tangent of A at p iff (1) A c~ t - - {p} (2) [A c~ g] > 2 for every other line g containing p.

Definition. Let P be a topological projective plane, A _c p. A line t is called a local tangent of A at p iff (1) p has a n e i g h b o r h o o d U with ( A n U) n t = {p} (2) t = lira p v a. a~A ct~ p

The definitions imply that tangents and local tangents, if they exist, are unique.

Definition. Let P be a projective plane, 0 _ P. 0 is called an oval iff (1) [ 0 n g l < 2 for each line g (2) 0 has a tangent at each point.

1-Dimensional Orbits in Flat Projective Planes

123

Definition. Let P be a topotogicaI projective plane, S~_P. S is called a

semiovaI iff (1) ]Sc~gl<2 for each line g (2) S has a local tangent t at each point p, and Sc~ t = {p}. (3) I S \ S 1 < 2 . S is called a spiral iff (1) Each point p has a neighborhood U such that I(S~ U ) ~ g l < 2 for each line g. (2) S has a local tangent in each point. (3) S = S u {p} w I for some line I and some point p~l. Theorem 2. Let P be a flat projective plane, A a subgroup of its automorphism group, A ~ $1 or R. Then every nontrivial orbit M of A has a local tangent at each point.

Proof. Let q~M. By Theorems 1.a, 1.b, q has a neighborhood U with Uc~ M = b ~ S 1 or ~ R and I b ~ g [ < 2 for each line g. Define b ' = b \ { q } and let f : P'-. {q} ~ oq be the continuous map q' ~ q v q'. The restriction of f to b' is injective and therefore a homeomorphism because of oq~S~. Define C = 0 for b ~ R and C = {c}, c~b', for b ~ S a. Then b'--. C is the union of two disjoint open intervals J~,J2 which both have q as one of their endpoints. Denote by t~ that endpoint of the open interval f(j~)~ ,q which corresponds to q for j~. For all t~[t~,t2].q..f~y)=J we have tc~b={q}. T h e o r e m 2 will be proved once we have shown ta = tz. Let l be a line fixed by A (Lemma 1). We claim that for each re J, b' is contained in one component of ( P \ l ) \ t . This is clear for b~S1. For b ~ R , assume that Jl and J2 are in different components C and C' of ( P \ l ) \ t. Select a q' from Jl and define m = q v q ' . Denote by D the component of ( P \ l ) \ m containing ]q', q[b, and by D' the other component. Then the two other components of b \ {q, q'} must be contained in D'. For otherwise, abbreviating u = l A m, and mapping b into u, by joining each of its points with u, we would find lines m'eu,, arbitrarily close to m, with [m'c~ bl > 3, a contradiction. Hence J2 is contained in C'c~ D'. For q" ~ ] q', q [b we have q" e C n D. Hence the component of (q" v q \ I) \ {q} not containing q" lies in C'nD'. Therefore f ( C ' ~ D ' ) lies in the component of . q \ {m, t} containing f(]q', q[b)- However, from the definition of q, t, t2 we have that this component is f(Jq', qEb)w [q, t [ ~,y~b'), and each of these lines is disjoint to J2, a contradiction. Assume now h~=t2. Select a t ~ J ' = J x { t a , t 2 } . Define x = t A l . There exists a symmetric neighborhood U of the identity in A and a neighborhood c ~ R of q in b such that xV~_{t'Allt'eJ '} and cV~_b. Then for all q'eq v we have x v q ' = t '~ for some t'EJ' and 6~ U. If q' is also in c, we have c~t'~ {q'}, and the corresponding property of t' shown in the preceding paragraph implies that c \ {q'} lies in one component of (P'-. I)--. x v q'. Now it is easy to see that

124

H.Groh: 1-Dimensional Orbits in Flat Projective Planes

for R ~ c ' _ P \ l and x ' e l there exist at most 2 lines n through x' such that [c'c~n]= 1 and c ' \ n is contained in one component of ( P \ l ) \ n . In our case (c'= c, x ' = x), however, c c~ qV is infinite, a contradiction. Putting b = M in the proof of Theorem 2, we have simultaneously proved: Lemma 6. Assume A < a P , and (a) A "~$1 or (b) A _~R and every fixed point of A lies on a fixed line. Then ]t ~ ML = 1 for every nontrivial orbit M and local tangent t of M. We summarize the geometric properties of 1-dimensional orbits in: Theorem 3. Let P be a flat projective plane, A a closed, connected, 1-dimensional subgroup of its automorphism group, and let M be a nontrivial orbit of A. Then one of the following cases holds: (a) A ~ Si, and M is an oval. (b) A ~_R, every fixed point of A lies on a fixed line, and M is a semioval. (c) A ~ R , there exists a fixed point p of A such that no line through p is fixed, and M is a spiral. Proof It is well known that $1 and R are the only 1-dimensional connected locally euclidean groups. For case (a), because of Theorem 1.a and Theorem 2, it suffices to show that for each q~M, the local tangent t at M is a tangent. This is clear from Lemma 6 and the proof of Theorem 2 by noting that for b = M ~ $1 we have . q = {q v m [ m E M \ {q}} w [ti, tZ].q..f(b, ) and that t l = t z = t . In case (b) it follows from Theorem 1.a, Theorem 2, Lemma 6 and Lemma 5 that M is a semioval. That M is a spiral in case (c) follows from Theorem 1.b, Theorem 2, and Lemma 5.

References 1. Alekseeva, G. P.: Topological classification of the collineations of the projective plane. Ivanov. gosudarst, ped. Inst., u~enye Zapiski, fiz.-mat. Nauki 5, 37-41 (1954) [M. Rev. 17, 998 (1954)]. 2. - Topological classification of collineations of the projective plane. Izvestija Vys~. U~ebn, Zaved. Mat. 1959, Nr. 2, 12-27 [M. Rev. 24, A461 (1959)]. 3. Brouwer, L.E.J.: Die Theorie der endlichen kontinuierlichen Gruppen, unabh~ingig von den Axiomen yon Lie. Math. Ann. 67, 246-267 (1909). 4. Hocking, J.G., Young, G.S.: Topology. Reading: Addison-Wesley 1961. 5. Salzmann, H.: Topologische projektive Ebenen. Math. Z. 67, 436-466 (1957). 6. - TopologischeStrukturzweidimensionalerprojektiverEbenen. Math. Z. 71, 408-413 (1959). 7. - Kompakte zweidimensionale projektive Ebenen. Math. Ann. 145, 401-428 (1962). 8. - Kompakte Ebenen mit einfacher Kollineationsgruppe. Arch. der Math. 13, 98-109 (1962). 9. - Zur Klassifikation topologischer Ebenen, Math. Ann. 150, 226-241 (1963). 10. Whyburn, G.T.: Topological analysis. Princeton: University Press 1958. Dr. Hansjoachim Groh Dept. of Mathematics Lakehead University Thunder Bay, Ontario Canada

(Received February 2, 1970)