arXiv subjects
Publications and source records attributed to Jianchun Wu.
The mapping torus induced by an automorphism $ϕ$ of the free abelian group $\mathbb{Z}^n$ is a semi-direct product $G=\mathbb{Z}^n\rtimes_ϕ\mathbb{Z}$. We show that whether the rank of $G$ is equal to $n+1$ is decidable. As a corollary, the rank of $\mathbb{Z}^3\rtimes_ϕ\mathbb{Z}$ is decidable.
We prove the pro-supersolvable closure of a finitely generated subgroup of the free group is finitely generated. It extends similar results for pro-$p$ closures proved by Ribes-Zalesskii and pro-Nilpotent closures proved by Margolis-Sapir-Weil.
We give an explicit characterization of which direct products $G$ of surface groups of Euclidean type satisfy that the fixed subgroup of any automorphism (or endomorphism) of $G$ is compressed, and of which is it always inert.
For a compact surface $Σ$ (orientable or not, and with boundary or not) we show that the fixed subgroup, $\operatorname{Fix} B$, of any family $B$ of endomorphisms of $π_1(Σ)$ is compressed in $π_1(Σ)$ i.e., $\operatorname{rk}((\operatorname{Fix} B)\cap H)\leq \operatorname{rk}(H)$ for any subgroup $\operatorname{Fix} B \leq H \leq π_1(Σ)$. On the way, we give a partial positive solution to the inertia conjecture, both for free and for surface groups. We also investigate direct products, $G$, of finitely many free and surface groups, and give a characterization of when $G$ satisfies that $\operatorname{rk}(\operatorname{Fix} ϕ) \leq \operatorname{rk}(G)$ for every $ϕ\in Aut(G)$.
For a closed surface $S$ with $χ(S)<0$, we show that the fixed subgroup of a family $\mathcal B$ of endomorphisms of $π_1(S)$ has $\rk \fix\mathcal B\leq \rk π_1(S)$. In particular, if $\mathcal B$ contains a non-epimorphic endomorphism, then $\rk \fix\mathcal B\leq \frac{1}{2} \rk π_1(S)$. We also show that geometric subgroups of $π_1(S)$ are inert, and hence the fixed subgroup of a family of epimorphisms of $π_1(S)$ is also inert.
For each closed oriented 3-manifold $M$ in Thurston's picture, the set of degrees of self-maps on $M$ is given.
Each closed oriented 3-manifold $M$ is naturally associated with a set of integers $D(M)$, the degrees of all self-maps on $M$. $D(M)$ is determined for each torus bundle and torus semi-bundle $M$. The structure of torus semi-bundle is studied in detail. The paper is a part of a project to determine $D(M)$ for all 3-manifolds in Thurston's picture.