ONE-DIMENSIONAL SUBGROUPS AND CONNECTED COMPONENTS IN NON-ABELIAN p-ADIC DEFINABLE GROUPS

WILLIAM JOHNSON, NINGYUAN YAO
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Abstract

We generalize two of our previous results on abelian definable groups in p-adically closed fields [12, 13] to the non-abelian case. First, we show that if G is a definable group that is not definably compact, then G has a one-dimensional definable subgroup which is not definably compact. This is a p-adic analogue of the Peterzil–Steinhorn theorem for o-minimal theories [16]. Second, we show that if G is a group definable over the standard model Abstract Image$\mathbb {Q}_p$, then Abstract Image$G^0 = G^{00}$. As an application, definably amenable groups over Abstract Image$\mathbb {Q}_p$ are open subgroups of algebraic groups, up to finite factors. We also prove that Abstract Image$G^0 = G^{00}$ when G is a definable subgroup of a linear algebraic group, over any model.

非阿贝尔 p-ADIC 可定义群中的一维子群和连接子群
我们将之前关于 p-adically 闭域中无差别可定义群的两个结果 [12, 13] 推广到非无差别情况。首先,我们证明,如果 G 是一个不可定义紧凑的可定义群,那么 G 有一个不可定义紧凑的一维可定义子群。这是 o 最小理论的 Peterzil-Steinhorn 定理的 p-adic 类似形式[16]。其次,我们证明,如果 G 是标准模型 $\mathbb {Q}_p$ 上的可定义群,那么 $G^0 = G^{00}$。作为应用,$\mathbb {Q}_p$ 上的可定义群是代数群的开放子群,直至有限因子。我们还证明,当 G 是线性代数群的可定义子群时,在任意模型上,$G^0 = G^{00}$ 。
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