arithmetic group造句

• Arithmetic groups can be used to construct isospectral manifolds.
• The hyperbolic triangle groups that are also arithmetic groups form a finite subset.
• One of the origins of the mathematical theory of arithmetic groups is algebraic number theory.
• Finally arithmetic groups are often used to construct interesting examples of locally symmetric Riemannian manifolds.
• The classical example of an arithmetic group is \ mathrm { SL } _ n ( \ mathbb Z ).
• This arithmetic construction can be generalised to obtain the notion of an " S-arithmetic group ".
• On the other hand, the class of groups thus obtained is not larger than the class of arithmetic groups as defined above.
• Another relevant list is that of K . Takeuchi, who classified the ( hyperbolic ) triangle groups that are arithmetic groups ( 85 examples ).
• The existence of congruence subgroups in an arithmetic groups provides it with a wealth of subgroups, in particular it shows that the group is residually finite.
• Let \ Gamma _ S be an S-arithmetic group in an algebraic group \ mathbf G \ subset \ mathrm { GL } _ d.
• The notion of an arithmetic group is a vast generalisation based upon the fundamental example of \ mathrm { SL } _ d ( \ mathbb Z ).
• Using this formula and certain number theoretic and Galois-cohomological estimates, Armand Borel and Gopal Prasad proved several finiteness theorems about arithmetic groups, [ 6 ].
• In higher dimensions the problem of finding the lattice of minimal volume is still open, though it has been solved when restricting to the subclass of arithmetic groups.
• The family of congruence subgroups in a given arithmetic group " & Gamma; " always has property ( & tau; ) of Lubotzky & ndash; Zimmer.
• Let \ Gamma be an arithmetic group : for simplicity it is better to suppose that \ Gamma \ subset \ mathrm { GL } _ n ( \ mathbb Z ).
• In particular there is a wealth of rigidity results in this setting, and a celebrated theorem of Grigori Margulis states that in most cases all lattices are obtained as arithmetic groups.
• The images of these representations are contained in arithmetic groups which are not symplectic, and this allows to construct many more finite quotients of \ mathrm { Mod } ( S ).
• An important question regarding the algebraic structure of arithmetic groups is the "'congruence subgroup problem "', which asks whether all subgroups of finite index are essentially congruence subgroups.
• Prasad and Rapinchuk introduced a new notion of " weak-commensurability " of arithmetic subgroups and determined " weak-commensurability classes " of arithmetic groups in a given semi-simple group.
• His research interests span the fields of Lie groups, their discrete subgroups, algebraic groups, arithmetic groups, geometry of locally symmetric spaces, and representation theory of reductive p-adic groups.