5 problems

Let $G$ and $H$ be linear sofic groups, let $G\ast H$ be their free product, and let $F_2$ denote the free group of rank $2$. Write $\mathcal{D}^{lin}_G(n)$ for the linear sofic pr…

Let $G$ be a finitely generated virtually nilpotent group, let $S$ be a finite generating set, and let $\mathcal{D}_{G,S}^{sof}$, $\mathcal{D}_{G,S}^{hyp}$,…

Let $H_{2l+1}$ denote the $(2l+1)$-dimensional Heisenberg group, and let $\mathcal{D}_{H_{2l+1},S}^{fin}(n)$ and $\mathcal{D}_{H_{2l+1},S}^{lin}(n)$ denote its weakly sofic and lin…

Let $G$ be a finitely generated virtually abelian group, and let $\operatorname{rank} G$ denote its rank. Write $\mathcal{D}_{G,S}^{lin}(n)$ for the linear sofic profile of $G$ rel…

Let $G$ be a sofic group. The sofic profile $athcal{D}_{G}^{sof}(n)$ and hyperlinear profile $athcal{D}_{G}^{hyp}(n)$ are compared using the relation $reccurlyeq$. The profile comp…