Droms RAAGs 上 $FP_n$ 型子直积
On subdirect products of type $FP_n$ of limit groups over Droms RAAGs
AI总结：
将自由群上极限群的结果推广到Droms RAAGs上，证明其子直积的有限指数性质并计算同调增长与$L^2$-贝蒂数。
摘要：
我们将自由群上的极限群和剩余自由群的一些已知结果分别推广到Droms RAAGs上的极限群和剩余Droms RAAGs。我们证明了Droms RAAGs上的极限群是自由群被无挠幂零群扩张的。我们证明，如果$S$是Droms RAAGs上具有平凡中心的极限群的$FP_s(\mathbb{Q})$型全子直积，那么$S$到任意$s$个Droms RAAGs上极限群直积的投影具有有限指数。此外，我们计算了Droms RAAGs上极限群在任意维数下的同调群增长和体积梯度，以及有限表示的剩余Droms RAAGs在$FP_m$型下维数不超过$m$时的相应量。特别地，这给出了这些群在相应维数下的解析$L^2$-贝蒂数的值。
英文摘要：
We generalize some known results for limit groups over free groups and residually free groups to limit groups over Droms RAAGs and residually Droms RAAGs, respectively. We show that limit groups over Droms RAAGs are free-by-(torsion-free nilpotent). We prove that if $S$ is a full subdirect product of type $FP_s(\mathbb{Q})$ of limit groups over Droms RAAGs with trivial center, then the projection of $S$ to the direct product of any $s$ of the limit groups over Droms RAAGs has finite index. Moreover, we compute the growth of homology groups and the volume gradients for limit groups over Droms RAAGs in any dimension and for finitely presented residually Droms RAAGs of type $FP_m$ in dimensions up to $m$. In particular, this gives the values of the analytic $L^2$-Betti numbers of these groups in the respective dimensions.
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