2026
Andre Nies, 30 Jan, The trivial units property and the unique product property for torsion free groups
Abstract: A torsion free group G satisfies the unique product property if for each pair A,B of finite nonempty subsets, some product in AB can be written uniquely. G satisfies the trivial units property over a domain R if the group algebra R[G] only has the trivial units, the ones of the form rg, where r is a unit in R and g is in G. Fixing a domain, the unique product property implies the trivial units property; the converse implication is not known.
Gardam in 2021 showed that GF_2[P] (where GF_2 is the two-element field) fails the trivial unit property for the Hantzsche-Wendt group P= < a,b | b^{-1} a^2 b = a^{-2}, a{-1}b^2 a = b^{-2}> . We will discuss the computational methods involving SAT solvers that Gardam used. Extending them, we found all 18 nontrivial units supported on the ball of radius 4 (the minimum possible value) in the Cayley graph of P.
We consider the Fibonacci groups F(n,n-1) where n \ge 4 is even. We show that each such group has a solvable word problem. We use SAT solvers to show that F(4,3) fails the unique product property, and discuss work in progress that might lead to a proof that it satisfies the trivial units property over GF_2.
This is joint work with Heiko Dietrich, Melissa Lee, and Marc Vinyals.