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Significance of flats in CAT(0) geometry : doctoral thesisZadnik, GašperŠtevilna 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].Type of material - dissertation ; adult, seriousPublication and manufacture - Ljubljana : [G. Zadnik], 2014Language - englishCOBISS.SI-ID - 16941401
Link(s):
http://www.matknjiz.si/doktorati/2014/Zadnik-14521-8.pdf
Repository of the University of Ljubljana – RUL
Digitalna knjižnica Slovenije - dLib.siDostop z namenskih računalnikov v prostorih NUK
Author
Zadnik, Gašper
Other authors
Smrekar, Jaka |
Caprace, Pierre-Emmanuel, 1981-
Topics
CAT(0) prostori |
grupa izometrij |
lokalno kompaktne grupe |
problem periodičnih ravnin |
CAT(0) spaces |
isometry group |
locally compact groups |
flat closing conjecture

Library/institution |
City | Acronym | For loan | Other holdings |
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FMF and IMFM, Mathematical Library, Ljubljana | Ljubljana | MAKLJ |
reading room 1 cop.
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National and University Library, Ljubljana | Ljubljana | NUK |
reading room 1 cop.
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not for loan 1 cop.
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Database name | Field | Year |
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Links to authors' personal bibliographies | Links to information on researchers in the SICRIS system |
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Zadnik, Gašper | 33290 |
Smrekar, Jaka | 21969 |
Caprace, Pierre-Emmanuel, 1981- | ![]() |
Source: Personal bibliographies
and: SICRIS
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