10 Poisson boundary
Problem 10.1 (Comments) [Vadim Kaimanovich] What is the Poisson boundary of the free group with an arbitrary measure?
Let
dvipng error! exitcode was 2 (signal 0), transscript follows:be a metric space and let
dvipng error! exitcode was 2 (signal 0), transscript follows:denote the space of continuous functions on
dvipng error! exitcode was 2 (signal 0), transscript follows:equipped with the topology of uniform convergence on bounded subsets. Fixing a basepoint
dvipng error! exitcode was 2 (signal 0), transscript follows:, the space
dvipng error! exitcode was 2 (signal 0), transscript follows:is continuously injected into
dvipng error! exitcode was 2 (signal 0), transscript follows:by
dvipng error! exitcode was 2 (signal 0), transscript follows:
If
dvipng error! exitcode was 2 (signal 0), transscript follows:is proper, then
dvipng error! exitcode was 2 (signal 0), transscript follows:is compact. The points on the boundary
dvipng error! exitcode was 2 (signal 0), transscript follows:are called horofunctions (or Busemann functions).
Problem 10.2 (Comments) [Conjecture of Anders Karlsson] There almost surely exists a horofunction
dvipng error! exitcode was 2 (signal 0), transscript follows:such that
dvipng error! exitcode was 2 (signal 0), transscript follows:
where
dvipng error! exitcode was 2 (signal 0), transscript follows:.
A theorem of Karlsson states that
dvipng error! exitcode was 2 (signal 0), transscript follows:there exists a horofunction
dvipng error! exitcode was 2 (signal 0), transscript follows:such that
dvipng error! exitcode was 2 (signal 0), transscript follows:
for all
dvipng error! exitcode was 2 (signal 0), transscript follows:.
Remark The problem above has been settled in the affirmative MR2271477.
Problem 10.3 (Comments) [Anders Karlsson] For any proper metric space it is possible to associate a kind of incidence geometry at infinity via horofunctions, halfspaces and their limits called stars. For the CAT(0) case, this structure is intimately connected with the Tits geometry, and for Teichm\"uller space it should relate well with the curve complex. In which situations do homomorphisms induce incidence preserving" maps between these geometries at infinity? Same problem for quasi-isometries.
Problem 10.4 (Comments) [Anders Karlsson] Consider the compactification of a finitely generated group constructed in the usual Stone-\v{C}ech way using the first
dvipng error! exitcode was 2 (signal 0), transscript follows:(or some other function space) cohomology. Is the associated incidence geometry at infinity always trivial (i.e., hyperbolic)? (This is related to problems of Gromov in Asymptotic invariants in the chapter on
dvipng error! exitcode was 2 (signal 0), transscript follows:cohomology.)
Kaimanovich and Masur proved the following: for a measure
dvipng error! exitcode was 2 (signal 0), transscript follows:on the mapping class group
dvipng error! exitcode was 2 (signal 0), transscript follows:(
dvipng error! exitcode was 2 (signal 0), transscript follows:can be any finite first moment, finite entropy probability measure such that the group generated by its support is non-elementary), there exists a measure
dvipng error! exitcode was 2 (signal 0), transscript follows:on
dvipng error! exitcode was 2 (signal 0), transscript follows:so that
dvipng error! exitcode was 2 (signal 0), transscript follows:
The measure
dvipng error! exitcode was 2 (signal 0), transscript follows:is called a
dvipng error! exitcode was 2 (signal 0), transscript follows:-stationary measure on
dvipng error! exitcode was 2 (signal 0), transscript follows:, this measure is unique. Here
dvipng error! exitcode was 2 (signal 0), transscript follows:is the Poisson boundary.
Problem 10.5 (Comments) [Moon Duchin] Characterize the hitting measure
dvipng error! exitcode was 2 (signal 0), transscript follows:on
dvipng error! exitcode was 2 (signal 0), transscript follows:obtained from the random walk by mapping classes on Teichm\"{u}ller space. Is it absolutely continuous with respect to visual measure (that is, Lebesgue measure on the visual sphere of directions)?
Problem 10.6 (Comments) [Moon Duchin] What is the Poisson boundary of Outer space?