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Higher modular groups as amalgamated products and a dichotomy for integral group rings. (arXiv:1811.12226v1 [math.GR])
来源于:arXiv
We give a concrete presentation for the general linear group defined over a
ring which is a finitely generated free $\mathbb{Z}$-module or the integral
Clifford group $\Gamma_n(\mathbb{Z})$ of invertible elements in the Clifford
algebra with integral coefficients. We then use this presentation to prove that
the elementary linear group over $\Gamma_n(\mathbb{Z})$ has a non-trivial
decomposition as a free product with amalgamated subgroup the elementary linear
group over $\Gamma_{n-1}(\mathbb{Z})$. This allows to obtain applications to
the unit group $\mathcal{U}(\mathbb{Z} G)$ of an integral group ring
$\mathbb{Z} G$ of a finite group $G$. In particular, we prove that $\mathcal{U}
(\mathbb{Z} G)$ is hereditary (FA), i.e. every subgroup of finite index has
property (FA), or is commensurable with a non-trivial amalgamated product. In
the case $\mathcal{U}(\mathbb{Z} G)$ is not hereditary (FA), we investigate
subgroups of finite index in $\mathcal{U}(\mathbb{Z} G)$ that have a
non-trivial 查看全文>>