We introduce and study the notion of a chain group of homeomorphisms of a one-manifold, which is a certain generalization of Thompson’s group F. The resulting class of groups exhibits a combination of uniformity and diversity. On the one hand, a chain group either has a simple commutator subgroup or the action of the group has a wandering interval. In the latter case, the chain group admits a c...

Consider two locally finite rooted trees as equivalent if each of them is a topological minor of the other, with an embedding preserving the tree-order. Answering a question of van der Holst, we prove that there are uncountably many equivalence classes.

We define a diffeomorphism invariant of smooth 4–manifolds which we can estimate for many smoothings of R and other smooth 4–manifolds. Using this invariant we can show that uncountably many smoothings of R support no Stein structure. Gompf [11] constructed uncountably many smoothings of R which do support Stein structures. Other applications of this invariant are given. AMS Classification numb...

C Croke and B Kleiner have constructed an example of a CAT(0) group with more than one visual boundary. J Wilson has proven that this same group has uncountably many distinct boundaries. In this article we prove that the knot group of any connected sum of two non-trivial torus knots also has uncountably many distinct CAT(0) boundaries.

We give a solution to the inverse problem (given prescribed function, find corresponding group) for large classes of speed, entropy, isoperimetric profile, return probability and $L_p$-compression functions finitely generated groups. For smaller classes, we solutions among solvable groups exponential volume growth. As corollaries, prove recent conjecture Amir on joint evaluation speed entropy e...