# Zero divisors with support size 3 in complex group algebras of residually finite groups

Conjecture. There exists a function $$f:\mathbb{N} \rightarrow \mathbb{N}$$ such that if $$\beta$$ is a non-zero element of the complex group algebra $$\mathbb{C}[G]$$ of a finite group $$G$$ such that $$1\in \text{supp}(\beta)$$, and $$(1+x+y) \cdot \beta =0$$ or $$(1+x-y) \cdot \beta =0$$ for some distinct and nontrivial $$x,y\in G$$, then $$\text{exp}\big (\langle x,y \rangle \big) \leq f(|\text{supp}(\beta)|)$$, where $$\text{exp}(H)$$ denotes the exponent of a finite group $$H$$ and $$\text{supp}(\gamma)$$ for an element $$\gamma=\sum_{g\in G} \gamma_g \; g \in \mathbb{C}[G]$$ denotes the set $$\{ g\in G \;|\; \gamma_g \not=0\}$$.

Motivation. If the conjecture is true, then the support of any zero divisor of the form $$1+x+y$$ or $$1+x-y$$ of the complex group algebra of any residually finite group generates a finite subgroup.

Proof. Let $$\alpha=1+x+y$$ or $$1+x-y$$ be a non-zero element of $$\mathbb{C}[G]$$ for some residually finite group $$G$$ such that $$\alpha \cdot \beta =0$$ for some non-zero $$\beta \in \mathbb{C}[G]$$. Let $$H=\langle \text{supp}(\alpha) \rangle$$. By Zelmanov's celebrated result on restricted Burnside problem, it is enough to show that the exponent of $$H$$ is finite. Since $$H$$ is finitely generated residually finite, there exists a descending series $$H=N_1\geq N_2 \geq \cdots$$ of normal subgroups $$H$$ of finite index such that $$\cap_{i\in\mathbb{N}} N_i=1$$. Note that there exists $$k\in\mathbb{N}$$ such that $$\bar{\alpha} \cdot \bar{\beta}=0$$ in $$\mathbb{C}[H/N_i]$$ for all $$i\geq k$$ and $$|\text{supp}(\bar{\beta})|=|\text{supp}(\beta)|=:t$$, where $$\bar{}$$ is the natural ring epimorphism from $$\mathbb{C}[H]$$ onto $$\mathbb{C}[H/N_i]$$. Now if the above conjecture is true then $$\text{exp}(H/N_i)\leq f(t)$$ and so $$\text{exp}(H)$$ is finite. This completes the proof.

If the above proof and conjecture are true then the integral group algebras of torsion-free residually finite groups have no zero divisor with support size $$3$$.