50 research outputs found
The boundary of the outer space of a free product
Let be a countable group that splits as a free product of groups of the
form , where is a finitely generated free
group. We identify the closure of the outer space
for the axes topology with the space of
projective minimal, \emph{very small} -trees, i.e. trees
whose arc stabilizers are either trivial, or cyclic, closed under taking roots,
and not conjugate into any of the 's, and whose tripod stabilizers are
trivial. Its topological dimension is equal to , and the boundary has
dimension . We also prove that any very small
-tree has at most orbits of branch points.Comment: v3: Final version, to appear in the Israel Journal of Mathematics.
Section 3, regarding the definition and properties of geometric trees, has
been rewritten to improve the exposition, following a referee's suggestio
A compactification of outer space which is an absolute retract
We define a new compactification of outer space (the \emph{Pacman
compactification}) which is an absolute retract, for which the boundary is a
-set. The classical compactification made of very small
-actions on -trees, however, fails to be locally -connected
as soon as . The Pacman compactification is a blow-up of
, obtained by assigning an orientation to every arc with
nontrivial stabilizer in the trees.Comment: Final version. To appear in Annales de l'Institut Fourie
Spectral theorems for random walks on mapping class groups and
We establish spectral theorems for random walks on mapping class groups of
connected, closed, oriented, hyperbolic surfaces, and on . In
both cases, we relate the asymptotics of the stretching factor of the
diffeomorphism/automorphism obtained at time of the random walk to the
Lyapunov exponent of the walk, which gives the typical growth rate of the
length of a curve -- or of a conjugacy class in -- under a random product
of diffeomorphisms/automorphisms.
In the mapping class group case, we first observe that the drift of the
random walk in the curve complex is also equal to the linear growth rate of the
translation lengths in this complex. By using a contraction property of typical
Teichm\"uller geodesics, we then lift the above fact to the realization of the
random walk on the Teichm\"uller space. For the case of , we
follow the same procedure with the free factor complex in place of the curve
complex, and the outer space in place of the Teichm\"uller space. A general
criterion is given for making the lifting argument possible.Comment: 45 pages, 3 figures. arXiv admin note: text overlap with
arXiv:1506.0724
Spectral rigidity for primitive elements of
Two trees in the boundary of outer space are said to be
\emph{primitive-equivalent} whenever their translation length functions are
equal in restriction to the set of primitive elements of . We give an
explicit description of this equivalence relation, showing in particular that
it is nontrivial. This question is motivated by our description of the
horoboundary of outer space for the Lipschitz metric. Along the proof, we
extend a theorem due to White about the Lipschitz metric on outer space to
trees in the boundary, showing that the infimal Lipschitz constant of an
-equivariant map between the metric completion of any two minimal, very
small -trees is equal to the supremal ratio between the translation
lengths of the elements of in these trees. We also provide approximation
results for trees in the boundary of outer space.Comment: 56 pages, 22 figue