東京大学大学院数理科学研究科
Graduate School of Mathematical Sciences, The University of Tokyo
Abstract
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.
発行年
2011-03-24
日本十進分類法
410
学位名
博士(数理科学)
学位
doctoral
学位分野
Mathematical Sciences (数理科学)
学位授与機関
University of Tokyo (東京大学)
研究科・専攻
Graduate School of Mathematical Sciences (数理科学研究科)
学位授与年月日
2011-03-24
学位授与番号
甲第27195号
学位記番号
博数理第376号
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Rigidity theorems for universal and symplectic universal lattices
MimuraM_23_3_PhD_a.pdf
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