AI 中文总结
该研究针对Feng Luo提出的紧3-流形基本群可通过有限交换环上PGL(2,R)群实现剩余有限的猜想，构造显式忠实表示证明其对六种Thurston几何成立，并给出两种几何的反例与修正条件。
AI 中文摘要
紧3-流形的基本群已知是剩余有限的。Feng Luo提出了一个更强的猜想：仅用形如$\text{PGL}(2,R)$的有限群即可实现这一性质，其中$R$是有限交换环。在前期工作中，该猜想已被证伪，不具有普遍成立性。这一猜想最初是在可几何化的可定向连通紧3-流形的语境下提出的。通过利用含幂零元的环构造显式忠实线性表示，我们证明该猜想对八种Thurston模型几何中的六种成立，即除$\text{S}^3$和$\text{SL}_2$的泛覆群$\tilde{\text{SL}_2}$之外的所有几何均成立。对于$\text{S}^3$几何，若将$\text{PGL}(2,R)$替换为$\text{GL}(2,R)$，则猜想成立；射影版本的球面反例是庞加莱同调球$\text{Σ}(2,3,5)$。对于$\tilde{\text{SL}_2}$几何，猜想的射影和非射影版本均不成立，Brieskorn球$\text{Σ}(2,3,7)$给出了一个反例。
英文摘要
The fundamental groups of compact 3-manifolds are known to be residually finite. Feng Luo conjectured that a stronger statement is true, by only allowing finite groups of the form $\mathrm{PGL}(2,R)$, where $R$ is a finite commutative ring. In earlier work, this conjecture was disproven in full generality. The conjecture arose in the context of orientable connected compact 3-manifolds which are geometrizable. By constructing explicit faithful linear representations using rings with nilpotent elements, we demonstrate that the conjecture holds for six of the eight Thurston model geometries, namely all but $\mathbb{S}^3$ and $\widetilde{\mathrm{SL}_2}$. In the case of $\mathbb{S}^3$, the conjecture holds if we replace $\mathrm{PGL}(2,R)$ with $\mathrm{GL}(2,R)$. A spherical counterexample for the projective variant is the Poincaré homology sphere $Σ(2,3,5)$. In the case of $\widetilde{\mathrm{SL}_2}$, the conjecture fails to hold for both the projective and non-projective variants; a counterxample is provided by the Brieskorn sphere $Σ(2,3,7)$.
Comments16 pages
