Asymptotic dimension
In
Formal definition
Let be a
We then define the asymptotic dimension as the smallest integer such that , if at least one such exists, and define otherwise.
Also, one says that a family of metric spaces satisfies uniformly if for every and every there exists a cover of by sets of diameter at most (independent of ) such that every closed -ball in intersects at most subsets from .
Examples
- If is a metric space of bounded diameter then .
- .
- .
- .
Properties
- If is a subspace of a metric space , then .
- For any metric spaces and one has .
- If then .
- If is a coarse embedding (e.g. a quasi-isometric embedding), then .
- If and are coarsely equivalent metric spaces (e.g. quasi-isometric metric spaces), then .
- If is a real tree then .
- Let be a Lipschitz map from a geodesic metric space to a metric space . Suppose that for every the set family satisfies the inequality uniformly. Then See[3]
- If is a metric space with then admits a coarse (uniform) embedding into a Hilbert space.[4]
- If is a metric space of bounded geometry with then admits a coarse embedding into a product of locally finite simplicial trees.[5]
Asymptotic dimension in geometric group theory
Asymptotic dimension achieved particular prominence in geometric group theory after a 1998 paper of Guoliang Yu[2] , which proved that if is a finitely generated group of finite homotopy type (that is with a classifying space of the homotopy type of a finite CW-complex) such that , then satisfies the Novikov conjecture. As was subsequently shown,[6] finitely generated groups with finite asymptotic dimension are topologically amenable, i.e. satisfy Guoliang Yu's Property A introduced in[7] and equivalent to the exactness of the reduced C*-algebra of the group.
- If is a word-hyperbolic groupthen .[8]
- If is relatively hyperbolicwith respect to subgroups each of which has finite asymptotic dimension then .[9]
- .
- If , where are finitely generated, then .
- For Thompson's group F we have since contains subgroups isomorphic to for arbitrarily large .
- If is the fundamental group of a finite graph of groups with underlying graph and finitely generated vertex groups, then[10]
- Mapping class groups of orientable finite type surfaces have finite asymptotic dimension.[11]
- Let be a connected Lie group and let be a finitely generated discrete subgroup. Then .[12]
- It is not known if has finite asymptotic dimension for .[13]
References
- ISBN 978-0-521-44680-8.
- ^ S2CID 17189763.
- MR 2231870.
- ISBN 978-0-8218-3332-2.
- MR 1928082.
- S2CID 250889716.
- S2CID 264199937.
- MR 2146189.
- S2CID 16743152.)
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: CS1 maint: unflagged free DOI (link - S2CID 14631642.
- S2CID 11350501.
- .
- MR 3286480. Ch. 9.1
Further reading
- Bell, Gregory; Dranishnikov, Alexander (2008). "Asymptotic dimension". .
- Buyalo, Sergei; Schroeder, Viktor (2007). Elements of Asymptotic Geometry. EMS Monographs in Mathematics. European Mathematical Society. ISBN 978-3-03719-036-4.