Thin Group Matrix Investigation

By Gabriel Konar-Steenberg and Michaela Polley

Some computational work related to a math research project undertaken with Dr. Eric Egge at Carleton College. The problem simplifies to this:

• Let A = [[1,1,2],[0,1,1],[0,-3,-2]] and B = [[-2,0,-1],[-5,1,-1],[3,0,1]].
• Let H be the set of all products of A and B: {A, B, AB, BA, AAB, ...}.
• Are there infinite left cosets of H within the set of all 3x3 matrices with determinant 1?

Most of the code here relates to examining what H consists of. For instance, we have that A^3 and B^3 are the identity matrix, so multiple strings of A and B can evaluate to the same matrix (e.g., AAAB = B). We call the shortest string that evaluates to a particular matrix its "reduced" form, but we can computationally verify that these are non-unique also: there are multiple strings of the same length that evaluate to the same matrix even when there are no shorter strings they can be reduced to (e.g. AABBABBAABAABB = BABABBABAABABA and both of those are reduced). reduction_investigation.py is a more-or-less brute force analysis that evaluates all possible reduced strings up to a given length, counts how many of them are actually reduced, and logs them for further analysis. A naïve approach would suggest that there are 2^n strings to be analyzed for string length n, but by using various optimization techniques and tools like Numba, we are able to complete this analysis for strings of length up to n=38 without specialized computing hardware in just a few hours.

The repository also contains code to analyze how elements of H behave modulo small integers, how the magnitude of elements of matrices in H changes as string length increases, and more.

This problem, first posed in Kontorovich et al. 2019 (who cite Long et al. 2011; see below) is related to the concept of thin groups and Zariski topology. Portions of this codebase also explore more general problems related to the one defined above.

References:

• Kontorovich, A., Long, D. D., Lubotzky, A., & Reid, A. W. (2019). WHAT IS...a Thin Group? Notices of the American Mathematical Society, 66(06), 1. https://doi.org/10.1090/noti1900
• Long, D. D., Reid, A. W., & Thistlethwaite, M. (2011). Zariski dense surface subgroups in SL(3,Z). Geometry & Topology, 15(1), 1–9. https://doi.org/10.2140/gt.2011.15.1