319 research outputs found
Lipschitz retraction and distortion for subgroups of Out(F_n)
Given a free factor A of the rank n free group F_n, we characterize when the
subgroup of Out(F_n) that stabilizes the conjugacy class of A is distorted in
Out(F_n). We also prove that the image of the natural embedding of Aut(F_{n-1})
in Aut(F_n) is nondistorted, that the stabilizer in Out(F_n) of the conjugacy
class of any free splitting of F_n is nondistorted, and we characterize when
the stabilizer of the conjugacy class of an arbitrary free factor system of F_n
is distorted. In all proofs of nondistortion, we prove the stronger statement
that the subgroup in question is a Lipschitz retract. As applications we
determine Dehn functions and automaticity for Out(F_n) and Aut(F_n).Comment: Version 3: 35 pages. Revised for publication. Changes from previous
versions: significant economies in exposition. Added an explicit description
of the stabilizer of a free splitting, in Lemma 1
Basic nets in the projective plane
The notion of basic net (called also basic polyhedron) on plays a
central role in Conway's approach to enumeration of knots and links in .
Drobotukhina applied this approach for links in using basic nets on
. By a result of Nakamoto, all basic nets on can be obtained from a
very explicit family of minimal basic nets (the nets , ,
in Conway's notation) by two local transformations. We prove a similar result
for basic nets in .
We prove also that a graph on is uniquely determined by its pull-back
on (the proof is based on Lefschetz fix point theorem).Comment: 14 pages, 15 figure
Line Patterns in Free Groups
We study line patterns in a free group by considering the topology of the
decomposition space, a quotient of the boundary at infinity of the free group
related to the line pattern. We show that the group of quasi-isometries
preserving a line pattern in a free group acts by isometries on a related space
if and only if there are no cut pairs in the decomposition space.Comment: 35 pages, 22 figures, PDFLatex; v2. finite index requires extra
hypothesis; v3. 37 pages, 24 figures: updated references and add example in
Section 6.3 of a rigid pattern for which the free group is not finite index
in the group of pattern preserving quasi-isometries; v4. 40 pages, 26
figures: improved exposition and add example in Section 6.4 of a rigid
pattern whose cube complex is not a tre
Detecting free splittings in relatively hyperbolic groups
We describe an algorithm which determines whether or not a group which is
hyperbolic relative to abelian groups admits a nontrivial splitting over a
finite group.Comment: 15 pages. Version 2 is 17 pages, edited in light of referee's
comments. To appear in Transactions of the AM
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