不可约格的升链、双可逆自动机与仿射算术群
Ascending chains of irreducible lattices, bi-reversible automata and affine arithmetic groups
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中文总结 AI 辅助
研究在\(n\)个齐次树乘积中构造不可约格升链，以及定义双可逆自动机使其定义的群有特定有限性长度，核心方法是借助全局函数域仿射群中\(S -\)算术群，贡献是实现相关构造。
中文摘要 AI 辅助
对于每个\(n \geq 2\)，我们在\(n\)个齐次树的乘积中构造一个不可约格的升链。此外，对于每对整数\(m_1, m_2 \geq 1\)，我们明确地定义一个双可逆自动机\(\mathcal B\)，使得由自动机\(\mathcal B\)定义的群\(G_{\mathcal B}\)具有有限性长度\(m_1\)，由对偶自动机定义的群\(G_{\mathcal B^*}\)具有有限性长度\(m_2\)。这两种构造都依赖于对全局函数域仿射群中\(S -\)算术群的考虑。
英文摘要
For each $n \geq 2$, we construct an ascending chain of irreducible lattices in the product of $n$ homogeneous trees. Moreover, for each pair of integers $m_1, m_2 \geq 1$, we define explicitly a bi-reversible automaton $\mathcal B$ such that the group $G_{\mathcal B}$ defined by the automaton $\mathcal B$ has finiteness length $m_1$ (i.e. it is of type $\mathrm{F}_{m_1}$ but not of type $\mathrm{FP}_{m_1+1}$), and the group $G_{\mathcal B^*}$ defined by the dual automaton has finiteness length $m_2$. Both constructions rely on the consideration of $S$-arithmetic groups in the affine group of a global function field.
