Tree Pairs for Algebraic Bieri-Strebel Groups

We reintroduce a previously discovered method for constructing tree pair representations for Algebraic Bieri-Strebel groups, as well as demonstrate a class of higher order groups that cannot have a tree pair representation. In doing so, we demonstrate that there is no maximum degree such that for all polynomials of higher degree, the associated Algebraic Bieri Strebel group must have a tree-pair representation.

💡 Research Summary

The paper revisits the construction of tree‑pair diagrams for a family of Thompson‑like groups known as algebraic Bieri‑Strebel groups, and then establishes a broad non‑existence result for such representations. It begins with a concise review of the classical tree‑pair model for Thompson’s group F, describing how each element can be encoded by a pair of rooted binary trees obtained from dyadic partitions of the unit interval. The authors explain the equivalence relation generated by inserting or deleting redundant carets and how composition of group elements corresponds to matching the right tree of one pair with the left tree of another. This machinery is then generalized to the Brown‑Thompson groups Fₙ, where each caret has n legs, reflecting the subdivision of intervals into n equal pieces.

Next, the paper introduces the Bieri‑Strebel construction G(I,A,P), where I is an interval, P a multiplicative subgroup of positive reals (the slope group), and A a Z