Here are a few problems that I think are nice.
- Find a finitely presented group with uncountably many non-homeomorphic (or even non -equivalent) asymptotic cones, independent of Continuum Hypothesis. Recall that if we
assume that CH is false than any uniform lattice in is an example (see Kramer, Shelah, Tent, Thomas, http://front.math.ucdavis.edu/0306.5420). The maximal number of non-homeomorphic asymptotic cones of a finitely generated group is continuum, there are finitely presented groups with two http://front.math.ucdavis.edu/0504.5350 and even infinitely many http://front.math.ucdavis.edu/1108.2100 non--equivalent asymptotic cones, and there are finitely generated groups with uncountably many non--equivalent asymptotic cones http://front.math.ucdavis.edu/0405.5030.
- Prove that every finitely generated group either has a simply connected asymptotic cone or an asymptotic cone with uncountable fundamental group. The history, motivation, and the main current results about this problem can be found in the papers by Curt Kent http://front.math.ucdavis.edu/1210.4420 and http://front.math.ucdavis.edu/1210.4411 .
- Define a “measure” on asymptotic cones of groups which will distinguish hyperbolic groups. It is known that all non-elementary hyperbolic groups have isometric asymptotic cones (the homogeneous -tree with every point of degree continuum. It would be nice to be able to distinguish asymptotic cones by introducing another piece of structure such as a “Brownian type” measure.
- Use asymptotic cones to define and study “true random Dehn function”. I will probably write about it in the next post.
- Prove or disprove that an amenable (non-virtually cyclic) group must have an asymptotic cone without cut-points. A candidate for a counterexample is in our paper on lacunary hyperbolic groups http://front.math.ucdavis.edu/0701.5365 (a finitely generated amenable non-virtually cyclic group with one asymptotic cone an -tree).