Fibonacci group

In mathematics, for a natural number [math]\displaystyle{ n \ge 2 }[/math], the nth Fibonacci group, denoted [math]\displaystyle{ F(2,n) }[/math] or sometimes [math]\displaystyle{ F(n) }[/math], is defined by n generators [math]\displaystyle{ a_1, a_2, \dots, a_n }[/math] and n relations:

• [math]\displaystyle{ a_1 a_2 = a_3, }[/math]
• [math]\displaystyle{ a_2 a_3 = a_4, }[/math]
• [math]\displaystyle{ \dots }[/math]
• [math]\displaystyle{ a_{n-2} a_{n-1} = a_n, }[/math]
• [math]\displaystyle{ a_{n-1}a_n = a_1, }[/math]
• [math]\displaystyle{ a_n a_1 = a_2 }[/math].

These groups were introduced by John Conway in 1965.

The group [math]\displaystyle{ F(2,n) }[/math] is of finite order for [math]\displaystyle{ n=2,3,4,5,7 }[/math] and infinite order for [math]\displaystyle{ n = 6 }[/math] and [math]\displaystyle{ n \ge 8 }[/math]. The infinitude of [math]\displaystyle{ F(2,9) }[/math] was proved by computer in 1990.

Kaplansky's unit conjecture

From a group [math]\displaystyle{ G }[/math] and a field [math]\displaystyle{ K }[/math] (or more generally a ring), the group ring [math]\displaystyle{ K[G] }[/math] is defined as the set of all finite formal [math]\displaystyle{ K }[/math]-linear combinations of elements of [math]\displaystyle{ G }[/math] − that is, an element [math]\displaystyle{ a }[/math] of [math]\displaystyle{ K[G] }[/math] is of the form [math]\displaystyle{ a = \sum_{g \in G} \lambda_g g }[/math], where [math]\displaystyle{ \lambda_g = 0 }[/math] for all but finitely many [math]\displaystyle{ g \in G }[/math] so that the linear combination is finite. The (size of the) support of an element [math]\displaystyle{ a = \sum\nolimits_g \lambda_g g }[/math] in [math]\displaystyle{ K[G] }[/math], denoted [math]\displaystyle{ |\operatorname{supp} a\,| }[/math], is the number of elements [math]\displaystyle{ g \in G }[/math] such that [math]\displaystyle{ \lambda_g \neq 0 }[/math], i.e. the number of terms in the linear combination. The ring structure of [math]\displaystyle{ K[G] }[/math] is the "obvious" one: the linear combinations are added "component-wise", i.e. as [math]\displaystyle{ \sum\nolimits_g \lambda_g g + \sum\nolimits_g \mu_g g = \sum\nolimits_g (\lambda_g \!+\! \mu_g) g }[/math], whose support is also finite, and multiplication is defined by [math]\displaystyle{ \left(\sum\nolimits_g \lambda_g g\right)\!\!\left(\sum\nolimits_h \mu_h h\right) = \sum\nolimits_{g,h} \lambda_g\mu_h \, gh }[/math], whose support is again finite, and which can be written in the form [math]\displaystyle{ \sum_{x \in G} \nu_x x }[/math] as [math]\displaystyle{ \sum_{x \in G}\Bigg(\sum_{g,h \in G \atop gh = x} \lambda_g\mu_h \!\Bigg) x }[/math].

Kaplansky's unit conjecture states that given a field [math]\displaystyle{ K }[/math] and a torsion-free group [math]\displaystyle{ G }[/math] (a group in which all non-identity elements have infinite order), the group ring [math]\displaystyle{ K[G] }[/math] does not contain any non-trivial units – that is, if [math]\displaystyle{ ab = 1 }[/math] in [math]\displaystyle{ K[G] }[/math] then [math]\displaystyle{ a = kg }[/math] for some [math]\displaystyle{ k \in K }[/math] and [math]\displaystyle{ g \in G }[/math]. Giles Gardam disproved this conjecture in February 2021 by providing a counterexample. He took [math]\displaystyle{ K = \mathbb{F}_2 }[/math], the finite field with two elements, and he took [math]\displaystyle{ G }[/math] to be the 6th Fibonacci group [math]\displaystyle{ F(2,6) }[/math]. The non-trivial unit [math]\displaystyle{ \alpha \in \mathbb{F}_2[F(2, 6)] }[/math] he discovered has [math]\displaystyle{ |\operatorname{supp} \alpha\,| = |\operatorname{supp} \alpha^{-1}| = 21 }[/math].

The 6th Fibonacci group [math]\displaystyle{ F(2,6) }[/math] has also been variously referred to as the Hantzsche-Wendt group, the Passman group, and the Promislow group.

External links

• An alternative proof that the Fibonacci group F(2,9) is infinite by Derek K. Holt (PostScript file).
• "Fibonacci group". https://encyclopediaofmath.org/wiki/Fibonacci_group.