论文标题
强的亚组复发和Nevo-Stuck-Zimmer定理
Strong subgroup recurrence and the Nevo-Stuck-Zimmer theorem
论文作者
论文摘要
令$γ$为一个可计数的群体，$ \ mathrm {sub}（γ）$它的chabauty空间，即紧凑型$γ$ - 空间，由所有子组组成$γ$。我们调用\ Mathrm {sub}（γ）$ boomerang子组的子组$δ\，如果每个$γ\中的每个$γ\，$γ^{n_i}δγ^{ - n_i} { - n_i} \ \ rightArrowδ$对于某些subsequence $ \ \ { Poincaré复发意味着$μ$ - 几乎每个子组$γ$都是回旋镖，就每个不变的随机子组$μ$ $γ$而言。我们为Boomerang子组建立许多与密度相关的特性，其中大多数几乎肯定对于不变的随机亚组而言。 令$ \ mathbb {k} $为一个数字字段，$ o $它的整数环，$ s $一组有限的估值，包括所有阿基米德估值，以及$ \ mathbb {g} $一个绝对几乎简单的群体，这些群体定义在$ \ m athbb {k} $上。我们的主要结果是，如果$ \ mathrm {rk} _ {\ mathbb {k}} \ Mathbb {g} \ ge 2 $，那么任何$ s $ s $ arithmetic grout $ \ mathbb {g}（g}（o__s）$都有很少的boomerang subgroomps subgrouff。也就是说，$γ$中的每个回旋镖都是有限和中央或有限指数。特别是，我们恢复了Margulis的正常亚组定理以及此类晶格的Nevo-Stuck-Zimmer定理。 我们在上述定理中包括一个简短的，可访问的证明，如果$γ$可与$ \ mathrm {sl} _n（\ mathbb {z}），\ n \ ge 3 $相称。
Let $Γ$ be a countable group and $\mathrm{Sub}(Γ)$ its Chabauty space, namely the compact $Γ$-space consisting of all subgroups of $Γ$. We call a subgroup $Δ\in \mathrm{Sub}(Γ)$ a boomerang subgroup if for every $γ\in Γ$, $γ^{n_i} Δγ^{-n_i} \rightarrow Δ$ for some subsequence $\{n_i \} \subset \mathbb{N}$. Poincaré recurrence implies that $μ$-almost every subgroup of $Γ$ is a boomerang, with respect to every invariant random subgroup $μ$ of $Γ$. We establish for boomerang subgroups many density related properties, most of which are known to hold almost surely for invariant random subgroups. Let $\mathbb{K}$ be a number field, $O$ its ring of integers, $S$ a finite set of valuations including all the Archimedean valuations, and $\mathbb{G}$ an absolutely almost simple group defined over $\mathbb{K}$. Our main result is that if $\mathrm{rk}_{\mathbb{K}} \mathbb{G} \ge 2$ then any $Γ$ which is commensurable to the $S$-arithmetic group $\mathbb{G}(O_S)$ has very few boomerang subgroups. Namely, every boomerang in $Γ$ is either finite and central or of finite index. In particular we recover Margulis' normal subgroup theorem as well as the Nevo-Stuck-Zimmer theorem for such lattices. We include a short, accessible proof for the above theorem in the case that $Γ$ is commensurable to $\mathrm{SL}_n(\mathbb{Z}), \ n \ge 3$.
