UTS Groups, Analysis, Geometry Seminar

Friday 30 January 2026, 11am-12noon Professor Andre Nies, University of Auckland, NZ

Title: The trivial units property and the unique product property for torsion free
groups 

Room: CB04.03.341 (Ground floor, building 4) 

All welcome! The webpage for the seminar series is https://sites.google.com/view/gaguts 

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.