Linear groups generated by reflection tori. (English) Zbl 0952.20042

Let \(V\) be a left vector space over a (possibly commutative) skew field \(k\). For \(g\in\text{GL}(V)\), define \([V,g]=\{vg-v\mid v\in V\}\) and \(C_V(g)=\{v\in V\mid vg-v=0\}\). The subspaces \([V,g]\) and \(C_V(g)\) are called center and axis of \(g\), respectively. If \(C_V(g)\) is a hyperplane and if \([V,g]\) is not contained in \(C_V(g)\), then \(g\) is called a reflection. Let \(H\) be a hyperplane and \(p\) some complement of \(H\) in \(V\). The group \(T_{p,H}\) consisting of the identity and all reflections with center \(p\) and axis \(H\) is called a reflection torus.
The authors determine all subgroups \(G\) of \(\text{GL}(V)\) generated by reflection tori.
If \(G\) is irreducible and if \(|k|>5\), then \(G=R(V,W^*)\), where \(W^*\) is a subspace of \(V^*\) with \(\text{Ann}(W^*)=0\); the group \(G\) contains a unique conjugacy class of reflection tori. Several reflection tori are described for \(|k|\leq 5\).


20H20 Other matrix groups over fields
51A50 Polar geometry, symplectic spaces, orthogonal spaces
20H25 Other matrix groups over rings
20E07 Subgroup theorems; subgroup growth
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