Deducing Theorem 4 from Theorem 3

The proposition above excludes triples of type I with λ=−1, and triples of type II with λ3 6= 0.

Hence, the proof of Theorem 3 is finished.

Remark IV.3.4. Here, we have found an important difference between the building and the smooth case: We have just shown (using the notation in- troduced in the previous chapter) that there are good S-sets (consisting of three points) which do not bound a convex rank 1-set.

For the real case, we will proceed to show: Let A be an S-set containing only triples corresponding to embedded hyperbolic planes. Then A bounds a convex rank 1-set (which may be chosen as a subset of a hyperbolic plane). It turns out that we do not need to assume that A is good (another contrast to the building case, see Example III.3.6).

IV.4

For_K=_R, we can classify all the possible asymptotic boundaries of convex rank 1-sets. Our argument uses the geometry of_RP2, so it does not generalize easily. Also, for_K=_C, one has to take more symmetric subspaces of rank 1 into account (there is also a copy of _RH3 _⊂_M

C).

In order to derive Theorem 4 from Theorem 3, it suffices to show: Proposition IV.4.1. Consider 4 points ηi, i ∈ {1,2,3,4} in ∂TMR, and suppose that each three of them are contained in the asymptotic boundary of a copy of _RH2. Then all ηi lie in the asymptotic boundary of the same copy

of _RH2_.

Proof. As usual, let each point ηi be the center of the Weyl chamber determined by the tuple (vi, pi) of incident 1- resp. 2-dimensional vector subspaces of _R3. Note that we may assume the ηi to be distinct (since otherwise, the claim is trivial).

We split the discussion in the following two cases:

• There are (at least) two triples of type II.

• Otherwise, there are (at least) three triples of type I. Let us discuss show the case of type II first:

Without loss of generality, we assume that (η1, η2, η3) and (η1, η2, η4) are

triples of type II. So hv1, v2, v3i = p is 2-dimensional, and since the ηi are distinct (and antipodal), p = hv1, v2i. By assumption, hv1, v2, v4i is 2-

dimensional, too, so it is equal to p.

Similarly, one derives that for i 6= j, pi ∩pj is independent from i, j, finishing the case of type II.

Now we consider the case of type I, and without loss of generality we assume that the triples (η1, η2, η3), (η1, η2, η4) and (η1, η3, η4) are of type I.

By our calculations above, this means that these triples lie on quadrics (in the sense described above) q4, q3, q2.

We know that q4, q3 meet tangentially in two points p1, p2 ∈RP2 (corre-

sponding to the Weyl chambers determined by η1, η2 resp.).

It is easy to see that the set of quadrics touching inp1, p2is a 1-dimensional

family; in particular, two quadrics touching in two points agree if they have a third point of intersection.

Observe that a quadric splits _RP2 _{into two connected components.}

It follows that if two distinct quadrics have two Weyl chambers in com- mon, then one of them lies entirely inside the other (except for the two points where they touch).

Now q4, q3, q2 each divideRP2 into two componentsAi, Bi. By the argu- ments above, we may assume A4 ⊂A3 ⊂A2 (and hence B2 ⊂B3 ⊂B4).

Ifq4 =q3, thenq2 touches this quadric in three points, soq2 =q4 =q3. So

we assume that theqi are pairwise distinct, and the inclusions named above are strict. In this case,q2 and q4 could only touch in the pointp1, not in two

points as required.

This contradiction finishes the proof.

Bibliography 85

Bibliography

[Ball95] Werner Ballmann. Lectures on spaces of nonpositive curvature. DMV Seminar, Band 25. Birkh¨auser, 1995.

[Bals06] Andreas Balser. Polygons with prescribed gauss map in hadamard spaces and euclidean buildings. Canadian Mathematical Bulletin, 49(3):321–336, 2006.

[BGS85] Werner Ballmann, Mikhael Gromov, and Viktor Schroeder. Man- ifolds of nonpositive curvature, volume 61 of Progress in Mathe- matics. Birkh¨auser Boston Inc., Boston, MA, 1985.

[BH99] Martin R. Bridson and Andr´e Haefliger. Metric spaces of nonpositive curvature. Springer, 1999.

[BL04] Andreas Balser and Alexander Lytchak. Building-like spaces, 2004, arXiv: math.MG/0410437. To appear in Journal of Math- ematics of Kyoto University.

[BL05] Andreas Balser and Alexander Lytchak. Centers of convex subsets of buildings. Ann. Global Anal. Geom., 28(2):201–209, 2005. [Ebe96] Patrick B. Eberlein. Geometry of nonpositively curved manifolds.

The University of Chicago Press, 1996.

[Gau63] C.F. Gauß. Letter to W. Bolyai. Werke, vol. 8, pages 220–225, 1863. Available from http://gdz.sub.uni-goettingen.de/en/. [Hel78] Sigurdur Helgason. Differential geometry, Lie groups, and sym-

metric spaces, volume 80 ofPure and Applied Mathematics. Aca- demic Press Inc. [Harcourt Brace Jovanovich Publishers], New York, 1978.

[HLS00] Christoph Hummel, Urs Lang, and Viktor Schroeder. Convex hulls in singular spaces of negative curvature. Ann. Global Anal. Geom., 18(2):191–204, 2000.

[Kar67] F. I. Karpeleviˇc. The geometry of geodesics and the eigen- functions of the Beltrami-Laplace operator on symmetric spaces.

Trans. Moscow Math. Soc., 1965:51–199. Amer. Math. Soc., Prov- idence, R.I., 1967, 1967.

[KL97] Bruce Kleiner and Bernhard Leeb. Rigidity of quasi-isometries for symmetric spaces and Euclidean buildings. Inst. Hautes ´Etudes Sci. Publ. Math., 86:115–197, 1997.

[KL06] B. Kleiner and B. Leeb. Rigidity of invariant convex sets in symmetric spaces. Inventiones Mathematicae, 163(3):657–676, 2006. [KLM04a] Michael Kapovich, Bernhard Leeb, and John J. Millson. Convex

functions on symmetric spaces, side lengths of polygons and the stability inequalities for weighted configurations at infinity, 2004,

arXiv:math.DG/0311486.

[KLM04b] Michael Kapovich, Bernhard Leeb, and John J. Millson. The generalized triangle inequalities in symmetric spaces and buildings with applications to algebra, 2004,arXiv:math.RT/0210256. To appear in Memoirs of the AMS.

[KLM04c] Michael Kapovich, Bernhard Leeb, and John J. Millson. Polygons in buildings and their refined side lengths, 2004,

arXiv:math.MG/0406305.

[Kre06] Robert Kremser. On buildings of non-archimedean norms from the perspective of non-positive curvature. Dipoma thesis, Univer- sity of Munich, 2006.

[Lee00] Bernhard Leeb. A characterization of irreducible symmetric spaces and Euclidean buildings of higher rank by their asymptotic

geometry. Bonner Mathematische Schriften [Bonn Mathemati-

cal Publications], 326. Universit¨at Bonn Mathematisches Institut, Bonn, 2000.

[Lee05] Bernhard Leeb. The geometry of SL(n,_R)/SO(n). Notes for a seminar in WS04/05, 2005.

[Lyt05] A. Lytchak. Rigidity of spherical buildings and joins. Geometric and Functional Analysis, 15(3):720–752, 2005.

Lebenslauf

1. Pers¨onliche Daten:

Name: Andreas Balser

Geburtsdatum: 28.09.1977

Geburtsort: Ichenhausen, Deutschland Familienstand: verheiratet, ein Sohn 2. Ausbildung:

09/84 - 06/97: Grundschule und Gymnasium in Gersthofen, Abitur-Note 1,0

07/97 - 07/98: Zivildienst (Arbeiter-Samariter-Bund Augsburg) 10/98 - 03/03: “Mathematics with Computer Science (MCS)”5-

Studium an der TU Darmstadt

Abschl¨usse: B.Sc. und Diplom mit Auszeichnung Diplomarbeit:

“On Trinoids and Minimal Disks bounded by Lines” 08/00 - 05/01: Auslandsstudium an der Tulane University, New Or-

leans, USA (Abschluss: M.Sc.)

04/03 - 10/06: Promotion in Differentialgeometrie bei Prof. Leeb, Ph.D., am mathematischen Institut der Ludwig-Maxi- milians-Universit¨at M¨unchen

5_{MCS ist ein international ausgerichteter Mathematik-Studiengang an der TU Darm-}

stadt mit Englisch als Unterrichtssprache im ersten Jahr. Inhaltlich handelt es sich um ein Mathematik-Studium mit Nebenfach Informatik.

In document Balser, Andreas (2006): On the Interplay between the Tits Boundary and the Interior of Hadamard Spaces. Dissertation, LMU München: Fakultät für Mathematik, Informatik und Statistik (Page 87-91)