Significance of flats in CAT(0) geometryZadnik, Gašper (Avtor) Smrekar, Jaka (Mentor) Caprace, Pierre-Emmanuel (Komentor) CAT(0) spacesisometry grouplocally compact groupsflat closing conjectureSeveral 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].[G. Zadnik]20142017-09-22 02:53:45Doktorsko delo/naloga95853UDK: 512.546(043.3)COBISS_ID: 16941401sl