2021-07-28T07:32:10Zhttps://repository.dl.itc.u-tokyo.ac.jp/oaioai:repository.dl.itc.u-tokyo.ac.jp:000040552021-03-01T19:48:43Z普遍格子と斜交普遍格子の剛性定理Rigidity theorems for universal and symplectic universal latticesMimura, Masato410University of Tokyo (東京大学)博士(数理科学)Cohomological rigidity theorems (with Banach coefficients) for some matrix groupsG over general rings are obtained. Main examples of these groups are (finite indexsubgroups of) universal lattices SLm(Z[x1, . . . , xk]) for m at least 3 and symplecticuniversal lattices Sp2m(Z[x1, . . . , xk]) for m at least 2 (where k is finite). The resultsincludes the following for certain large m:(1) The first group cohomology vanishing with any isometric Lp or p-Schatten coefficients,where p is any real on (1,∞). This is strictly stronger than havingKazhdan’s property (T).(2) The injectivity of the comparison map in degree 2 from bounded to ordinarycohomology, with coefficients as in item (1) not containing trivial one.As a corollary, homomorphim rigidity (, namely, the statement that every homomorphismfrom G has finite image) is established with the following targets: circlediffeomorhisms with low regularity; mapping class groups of surfaces; and outerautomorhisms of free groups. These results can be regarded as a generalization ofsome previously known rigidity theorems for higher rank lattices (Bader–Furman–Gelander–Monod; Burger–Monod; Farb–Kaimanovich–Masur; Bridson–Wade) tothe case of certain general matrix group cases, which are not realizable as lattices inalgebraic groups. Note that G above does not usually satisfy the Margulis finitenessproperty.Finally, quasi-homomorphims are studied on special linear groups over euclideandomains. This concept has relation to item (2) above for trivial coefficient case,and to the conception of the stable commutator length. In particular, a question ofM. Ab´ert and N. Monod, which was for instance stated at ICM 2006, is answeredfor large degree case, and a new example of groups with the following intriguingfeatures is provided: having infinite commutator width; but the stable commutatorlength vanishing.thesis2011-03-242011-03-24application/pdfapplication/pdf甲第27195号https://repository.dl.itc.u-tokyo.ac.jp/record/4055/files/MimuraM_23_3_PhD_a.pdfhttps://repository.dl.itc.u-tokyo.ac.jp/record/4055/files/MimuraM_23_3_PhD_b.pdfeng