AI 中文总结
研究离散群的虚拟满射及相关猜想，通过证明关于性质$F_{n}$和$FP_{n}$的虚拟满射猜想，在特定假设下证明同调$n-(n + 1)-(n + 2)$猜想，进而推导出性质$F_{n}$的$n-(n + 1)-(n + 2)$猜想。
AI 中文摘要
我们证明了离散群关于性质$F_{n}$和性质$FP_{n}$的虚拟满射猜想：给定$F_k$（分别为$FP_k$）型群的乘积，一个几乎满射到$k$元组的子群也必定是$F_k$（分别为$FP_k$）。在公共商是有限表示的假设下，我们证明了离散群的同调$n-(n + 1)-(n + 2)$猜想，并证明该假设不能去掉。我们由此推导出性质$F_{n}$的$n-(n + 1)-(n + 2)$猜想。
英文摘要
We prove the Virtual Surjection Conjecture for discrete groups, both for property $F_{n}$ and for property $FP_{n}$: given a product of groups of type $F_k$ (respectively $FP_k$), a subgroup that virtually surjects onto $k$-tuples must be $F_k$ (respectively $FP_k$) as well. We prove the homological $n$-$(n+1)$-$(n+2)$ Conjecture for discrete groups under the assumption the common quotient is finitely presented, and prove that this assumption cannot be dropped. We deduce the $n$-$(n+1)$-$(n+2)$ Conjecture for property $F_{n}$.
