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On the dimension growth of groups
Dimension growth functions of groups have been introduced by Gromov in 1999.
We prove that every solvable finitely generated subgroups of the R. Thompson
group has polynomial dimension growth while the group itself, and some
solvable groups of class 3 have exponential dimension growth with exponential
control. We describe connections between dimension growth, expansion properties
of finite graphs and the Ramsey theory.Comment: 20 pages; v3: Erratum and addendum included as Section 9. We can only
prove that the lower bound of the dimension growth of is exp sqrt(n). New
open questions and comments are added. v4: The paper is completely revised.
Dimension growth with control is introduced, connections with graph expansion
and Ramsey theory are include
Thick hyperbolic 3-manifolds with bounded rank
We construct a geometric decomposition for the convex core of a thick
hyperbolic 3-manifold M with bounded rank. Corollaries include upper bounds in
terms of rank and injectivity radius on the Heegaard genus of M and on the
radius of any embedded ball in the convex core of M.Comment: 170 pages, 17 figure
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