论文标题
直接产品的二进制亚组
Binary Subgroups of Direct Products
论文作者
论文摘要
我们探索了一种基本结构，该结构产生有限呈现的群体，具有不同的同源性属性 - {\ em二进制亚组}，$ b（σ，μ）<g_1 \ times \ times \ dots \ dots \ times g_m $。这些完整的细分产品需要很少的发电机。如果每个$ g_i $有限地呈现，则有限地呈现$ b（σ，μ）$。当$ g_i $是非亚伯限制组（例如自由或表面组）时，$ b（σ，μ）$提供了有限呈现的，没有有限分类空间的新示例，而不是Stallings-bieri类型。这些例子解决了Minasyan的问题，该问题将不同的无剩余组的等级概念联系起来。使用二进制亚组，我们证明，如果$ g_1，\ dots，g_m $是完美的组，每个组最多需要$ r $发电机，则$ g_1 \ times \ dots \ times \ times g_m $最多需要$ rfloor \ log_2 m+log_2 m+1 \ rfloor $ $生成器。
We explore an elementary construction that produces finitely presented groups with diverse homological finiteness properties -- the {\em binary subgroups}, $B(Σ,μ)<G_1\times\dots\times G_m$. These full subdirect products require strikingly few generators. If each $G_i$ is finitely presented, $B(Σ,μ)$ is finitely presented. When the $G_i$ are non-abelian limit groups (e.g. free or surface groups), the $B(Σ,μ)$ provide new examples of finitely presented, residually-free groups that do not have finite classifying spaces and are not of Stallings-Bieri type. These examples settle a question of Minasyan relating different notions of rank for residually-free groups. Using binary subgroups, we prove that if $G_1,\dots,G_m$ are perfect groups, each requiring at most $r$ generators, then $G_1\times\dots\times G_m$ requires at most $r \lfloor \log_2 m+1 \rfloor$ generators.
