For example, Theorem 3.2 is used in [SZ2] to obtain symplectic versions of the following results in complex geometry: • the asymptotic expansion theorem of [Ze1], • the Tian almost isometry theorem [Ti], • the Kodaira embedding theorem (see [GH] or [SSo]).
Universality and scaling of zeros on symplectic manifolds
This results in an inequality between the Hamiltonian and the action per isometry period, the difference being them being the entropy.
Generalized entropy and Noether charge
Note that the quasi-isometry type of a metric space X is unchanged upon removal of any bounded subset of X ; hence the term “asymptotic”.
Problems on the geometry of finitely generated solvable groups
The word metric on any f.g. group is unique up to quasi-isometry. 2.
Problems on the geometry of finitely generated solvable groups
The set of self quasi-isometries of a space X , with the operation of composition, becomes a group QI(X ) once one mods out by the relation f ∼ g if d(f , g) < ∞ in the sup norm.
Problems on the geometry of finitely generated solvable groups
There are many other quasi-isometry invariants for ﬁnitely-generated nilpotent groups Γ.
Problems on the geometry of finitely generated solvable groups
It is not known whether or not the Malcev completion is a quasi-isometry invariant.
Problems on the geometry of finitely generated solvable groups
The numbers ci (Γ) are quasi-isometry invariants.
Problems on the geometry of finitely generated solvable groups
In particular we recover (as special cases) that growth and cohomological dimension are quasi-isometry invariants of Γ.
Problems on the geometry of finitely generated solvable groups
K, 0) quasi-isometry, i.e. is a bilipschitz homeomorphism.
Problems on the geometry of finitely generated solvable groups
It is then shown that “quasi-isometries remember the dynamics”.
Problems on the geometry of finitely generated solvable groups
If there is a horizontalrespecting quasi-isometry f : GM → GN then there exist nonzero a, b ∈ R so that M a and M b have the same absolute Jordan form.
Problems on the geometry of finitely generated solvable groups
While we have already seen that there is a somewhat ﬁne classiﬁcation of ﬁnitely presented, nonpolycyclic abelian-by-cyclic groups up to quasi-isometry, this class of groups is but a very special class of ﬁnitely generated solvable groups.
Problems on the geometry of finitely generated solvable groups
Farb, The quasi-isometry classiﬁcation of lattices in semisimple Lie groups, Math.
Problems on the geometry of finitely generated solvable groups
Whyte, Quasi-isometries and groups acting on trees, in preparation. P.
Problems on the geometry of finitely generated solvable groups
***