Significance of flats in CAT(0) geometry : doctoral thesis

Zadnik, Gašper (author), Smrekar, Jaka (mentor), Caprace, Pierre-Emmanuel (comentor)

Several questions/conjectures in CAT(0) geometry are inspired by analogous theorems that are known to hold for Riemannian manifolds of nonpositive sectional curvature. This thesis deals with the one which was settled by Bangert and Schröder in early nineties for real analytic manifolds, [V Bangert, v Schröder, Existence of flat tori in analytic manifolds of nonpositive curvature. Ann. Sci. École Norm. Sup. 24 (1992), no. 4 pp. 605-634]. It is called the flat closing problem and it predicts a copy of ▫$\mathbb{Z}^m$▫ in any discrete group which acts properly and cocompactly by isometries on a CAT(0) space ▫$X$▫ containing an isometric copy of ▫$\mathbb{R}^m$▫. We summarize results from [P.-E. Caprace, N. Monod, Isometry groups of non-positively curved spaces: structure theory. J. Topol. 2 (2009), no. 4, pp. 661-700 and P.-E. Caprace, N. Monod, Isometry groups of non-positively curved spaces: discrete subgroups. J. Topol. 2 (2009), no. 4, pp. 701-746] about the full isometry group of a proper, cocompact and geodesically complete CAT(0) space. Then we apply those results to prove the main theorem from [P.-E. Caprace, G. Zadnik, Regular elements in CAT(0) groups. Preprint at http://arXiv.org/abs/1112.4637 (2011)], a very partial answer to the flat closing conjecture: "If a proper CAT(0) space ▫$X$▫ is a product of ▫$m$▫ geodesically complete factors, then discrete ▫$\Gamma$▫, which acts properly and cocompactly on ▫$X$▫, contains a copy of▫ $\mathbb{Z}^m$▫." Even though the theorem above is far from the full generality of the flat closing problem, its proof uses a deep machinery from the structure theory of the isometry group of the corresponding CAT(0) space. The proof relies in an essential way to the solution of Hilbert's fifth problem (Theorem Glaeson, Montgomery-Zippin). This solution leads to a dichotomy for the isometry group of a nice non Euclidean CAT(0) space - either it is a Lie group or a totally disconnected locally compact group. Applying this dichotomy to the irreducible factors from the theorem, we deal with two separated approaches. The first case is covered by older results from Lie group theory while the second relies to the geometric properties of CAT(0) space with totally disconnected isometry group, see [P.-E. Caprace, N. Monod, Isometry groups of non-positively curved spaces: structure theory. J. Topol. 2 (2009), no. 4, pp. 661-700].

• Language: English
• Keywords: CAT(0) spaces, isometry group, locally compact groups, flat closing conjecture
• Work type: Doctoral dissertation
• Typology: 2.08 - Doctoral Dissertation
• Organization: FMF - Faculty of Mathematics and Physics
• Place of publishing: Ljubljana
• Publisher: [G. Zadnik]
• Year: 2014
• Number of pages: 46 str.
• PID: 20.500.12556/RUL-95853
• UDC: 512.546(043.3)
• COBISS.SI-ID: 16941401
• Publication date in RUL: 24.10.2017

Secondary language

• Language: Slovenian
• Title: Pomen ravnin v CAT(0) geometriji

Abstract

Številna vprašanja v CAT(0) geometriji izvirajo iz izrekov o Riemannovih mnogoterostih nepozitivnih prereznih ukrivljenosti. V disertaciji se ukvarjamo z enim izmed njih, s problemom periodičnih ravnin. V kontekstu realnih analitičnih mnogoterosti sta ga rešila Bangert in Schröder, [V Bangert, v Schröder, Existence of flat tori in analytic manifolds of nonpositive curvature. Ann. Sci. École Norm. Sup. 24 (1992), no. 4 pp. 605-634]. Problem sprašuje, ali vedno lahko najdemo kopijo proste abelove grupe ▫$\mathbb{Z}^m$▫ v grupi, ki deluje kokompaktno diskretno z izometrijami na CAT(0) prostoru ▫$X$▫, ki vsebuje izometrično vloženo kopijo ▫$\mathbb{R}^m$▫. V uvodnih poglavjih povzamemo dognanja iz del [P.-E. Caprace, N. Monod, Isometry groups of non-positively curved spaces: structure theory. J. Topol. 2 (2009), no. 4, pp. 661-700 in P.-E. Caprace, N. Monod, Isometry groups of non-positively curved spaces: discrete subgroups. J. Topol. 2 (2009), no. 4, pp. 701-746] o celotni grupi izometrij pravega kokompaktnega geodezično polnega CAT(0) prostora. Nato ta dognanja uporabimo v dokazu glavnega izreka iz [P.-E. Caprace, G. Zadnik, Regular elements in CAT(0) groups. Preprint at http://arXiv.org/abs/1112.4637 (2011)], ki poda delni odgovor na problem periodičnih ravnin: "Naj bo parvi CAT(0) prostor ▫$X$▫ produkt ▫$m$▫ geodezično polnih faktorjev. Tedaj poljubna grupa ▫$\Gamma$▫, ki deluje kokompaktno diskretno z izometrijami na ▫$X$▫, vsebuje kopijo ▫$\mathbb{Z}^m$▫." Čeprav predpostavke zapisanega izreka močno posežejo v splošnost problema periodičnih ravnin, so za njegov dokaz potrebni globoki izreki iz strukturne teorije grupe izometrij dotičnega CAT(0) prostora. Za dokaz ključna je rešitev Hilbertovega petega problema (izrek Glaeson, Montgomery-Zippin), ki zagotavlja dihotomojo za grupe izometrij določenih CAT(0) prostorov. Bodisi je grupa izometrij Liejeva bodisi je popolnoma nepovezana lokalno kompaktna topološka grupa. Glede na to dihotomijo se dokaz izreka razdeli na dva dela. Prvi del sledi iz znanih izrekov iz teorije Liejevih grup, med tem ko se drugi del sklicuje na geometrijo CAT(0) prostora s popolnoma nepovezano grupo izometrij, [P.-E. Caprace, N. Monod, Isometry groups of non-positively curved spaces: structure theory. J. Topol. 2 (2009), no. 4, pp. 661-700].

• Keywords: CAT(0) prostori, grupa izometrij, lokalno kompaktne grupe, problem periodičnih ravnin