93 research outputs found
Spheres and Projections for
The outer automorphism group Out(F_2g) of a free group on 2g generators
naturally contains the mapping class group of a punctured surface as a
subgroup. We define a subsurface projection of the sphere complex of the
connected sum of n copies of S^1 x S^2 into the arc complex of the surface and
use this to show that this subgroup is a Lipschitz retract of Out(F_2g). We
also use subsurface projections to give a simple proof of a result of Handel
and Mosher [HM10] stating that stabilizers of conjugacy classes of free
splittings and corank 1 free factors in a free group Fn are Lipschitz retracts
of Out(F_n).Comment: 32 pages, 9 figure
Characteristic varieties of graph manifolds and quasi-projectivity of fundamental groups of algebraic links
The present paper studies the structure of characteristic varieties of
fundamental groups of graph manifolds. As a consequence, a simple proof of
Papadima's question is provided on the characterization of algebraic links that
have quasi-projective fundamental groups. The type of quasi-projective
obstructions used here are in the spirit of Papadima's original work.Comment: 22 pages, 6 figures, to appear in European Journal of Mathematic
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
Between buildings and free factor complexes: A Cohen-Macaulay complex for Out(RAAGs)
For every finite graph , we define a simplicial complex associated to
the outer automorphism group of the RAAG . These complexes are
defined as coset complexes of parabolic subgroups of and
interpolate between Tits buildings and free factor complexes. We show that each
of these complexes is homotopy Cohen-Macaulay and in particular homotopy
equivalent to a wedge of d-spheres. The dimension d can be read off from the
defining graph and is determined by the rank of a certain Coxeter
subgroup of . In order to show this, we refine the
decomposition sequence for established by Day-Wade,
generalise a result of Brown concerning the behaviour of coset posets under
short exact sequences and determine the homotopy type of relative free factor
complexes associated to Fouxe-Rabinovitch groups.Comment: 56 pages, 5 figure
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