Fall 2021
Students: Simon Xiang, Jimmy Xin, Ruiqi Zou
Faculty advisor: Lvzhou (Joe) Chen

The stable commutator length (scl) is a characteristic (length-)function on groups, which carries geometric and dynamical information. It has various applications in group theory such as Gromov's problem about surface subgroups and homomorphism rigidities. See the background section for the definition of scl and its basic properties.

A groundbreaking theorem of Calegari shows that scl is rational in free groups and can be computed by an efficient algorithm. However, it is difficult to understand scl for a family of words in practice, despite of some known examples and results.

The aim of this project is to explore scl in nice families of words in free groups and discover explicit formulas via computer experiments. This problem is related to several conjectures about the spectrum of scl (i.e. the set of scl values) in free groups.

As an illustration, one can observe scl of the following sequence of words via scallop-master (an algorithm that computes scl) in the free group generated by a,b (with A=a^-1 and B=b^-1).

\(\mathrm{scl}(abaaaBAAAbABabAB)=1=6/6\\ \mathrm{scl}(abaaaaBAAAAbABabAB)=9/8\\ \mathrm{scl}(abaaaaaBAAAAAbABabAB)=6/5=12/10\\ \mathrm{scl}(abaaaaaaBAAAAAAbABabAB)=5/4=15/12\\ \mathrm{scl}(abaaaaaaaBAAAAAAAbABabAB)=9/7=18/14\)

By taking suitable denominators as above, it is reasonable to guess the following formula

\(\mathrm{scl}(aba^n B A^n bABabAB)=\frac{3n-3}{2n},\)

which limits to 3/2 as n goes to infinity.