Non-rigid regions of real Grothendieck groups of gentle and special biserial algebras

In the representation theory of finite-dimensional algebras $A$ over a field, the classification of 2-term (pre)silting complexes is an important problem. One of the useful tool is the g-vector cones associated to the 2-term presilting complexes in the real Grothendieck group $K_0(\operatorname{$\mathsf{p}\mathsf{r}\mathsf{o}\mathsf{j}$} A){$\mathbb{R}$}:=K_0(\operatorname{$\mathsf{p}\mathsf{r}\mathsf{o}\mathsf{j}$} A) \otimes{$\mathbb{Z}$} {$\mathbb{R}$}$\mathrm{.}Theaimofthispaperistostudythecomplement$\operatorname{$\mathsf{N}\mathsf{R}$}$oftheunion$\operatorname{$\mathsf{C}\mathsf{o}\mathsf{n}\mathsf{e}$}$ of all g-vector cones, which we call the non-rigid region. By the work of Iyama and us, $\mathrm{}$ is determined by 2-term presilting complexes and a certain closed subset $R_0 \subset K_0(\operatorname{$\mathsf{p}\mathsf{r}\mathsf{o}\mathsf{j}$} A)_{$\mathbb{R}$}$, which is called the purely non-rigid region. In this paper, we give an explicit description of $R_0$ for complete special biserial algebras in terms of a finite set of maximal nonzero paths in the Gabriel quiver of $A$. We also prove that $\mathrm{}$ has some kind of fractal property and that $\mathrm{}$ is contained in a union of countably many hyperplanes of codimension one. Thus, any complete special biserial algebra is g-tame, that is, $\mathrm{}$ is dense in $K_0(\mathrm{}A{)}_{\mathbb{R}}$.