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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-termpresilting complexes in the real Grothendieck group$K_0(\operatorname{\mathsf{proj}}A)_{\mathbb{R}}:=K_0(\operatorname{\mathsf{proj}} A) \otimes_{\mathbb{Z}}{\mathbb{R}}$. The aim of this paper is to study the complement$\operatorname{\mathsf{NR}}$ of the union $\operatorname{\mathsf{Cone}}$ of allg-vector cones, which we call the non-rigid region. By the work of Iyama andus, $\operatorname{\mathsf{NR}}$ is determined by 2-term presilting complexesand a certain closed subset $R_0 \subset K_0(\operatorname{\mathsf{proj}}A)_{\mathbb{R}}$, which is called the purely non-rigid region. In this paper,we give an explicit description of $R_0$ for complete special biserial algebrasin terms of a finite set of maximal nonzero paths in the Gabriel quiver of $A$.We also prove that $\operatorname{\mathsf{NR}}$ has some kind of fractalproperty and that $\operatorname{\mathsf{NR}}$ is contained in a union ofcountably many hyperplanes of codimension one. Thus, any complete specialbiserial algebra is g-tame, that is, $\operatorname{\mathsf{Cone}}$ is dense in$K_0(\operatorname{\mathsf{proj}} A)_{\mathbb{R}}$.

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