It's a well-known result due to J. Tits that a finite-dimensional real reflection group has a faithful presentation, given by its Coxeter diagram (i.e. the linear group in question is isomorphic to the corresponding finitely presented abstract group). In our setting, we have an infinite subgroup $H\le \operatorname{GL}(n)$ generated by finitely many complex involutory reflections, i.e. transformations that have the spectrum $(1,1,\dotsc,1,-1)$.

What can be said about a presentation for $H$? Does $H$ at least admit a finite presentation? If it matters, only one of our involutory generators has non-real entries.

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    $\begingroup$ It's not clear what you mean by "faithful presentation" and "finite faithful presentation". In any case, the isomorphism between a finite real reflection group and the abstract group given by the Coxeter presentation is due to Coxeter and not Tits. There is an unrelated notion of "faithful linear representation" (finite dimensional or otherwise) of an abstractly described group. By the way, the tag "invariant-theory" doesn't seem useful here. $\endgroup$ Jan 14, 2012 at 16:15
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    $\begingroup$ Do you mean $U(n)$? $\endgroup$
    – Ian Agol
    Jan 14, 2012 at 20:56
  • $\begingroup$ Jim, our $H$ are not finite. The question is whether $H$ is finitely presented. I'll edit accordingly. $\endgroup$ Jan 15, 2012 at 14:41
  • $\begingroup$ PS. "faithful representation" - a representation of $H$ without a kernel, whereas "faithful presentation" - a presentation (i.e. a surjective morphism from an f.p. group to $H$) without a kernel. OK, maybe this is not the standard way of taking about these things in English-language literature... Thus I removed the word "faithful" from the question. $\endgroup$ Jan 15, 2012 at 14:57
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    $\begingroup$ HW, the group here is not discrete (hence not a CRG), so even in the real case the question is nontrivial and the answer doesn't immediately follow from the theory of Coxeter groups. $\endgroup$ Jan 16, 2012 at 4:44

1 Answer 1


The answer is almost surely negative. Below are two examples of discrete non-finitely presentable subgroups of $GL(n,{\mathbb C})$ which are generated by finitely many finite order elements. In the first example generators are involutions but some have more than one eigenvalues $-1$, in the second example generators are "complex reflections," of order $5$, but where all but one eigenvalues are equal to $1$. By working harder one should be able to construct examples you are asking for, but I do not see sufficient motivation for doing so. My guess is that you were hoping for a positive result instead of counter-examples.

Example 1. This example is a variation on the example in

[1] M.Feighn and G. Mess, Conjugacy Classes of Finite Subgroups of Kleinian Groups, American Journal of Mathematics, Vol. 113, No. 1 (1991), 179-188. https://doi.org/10.2307/2374827

See also:

[2] G. Baumslag and J. E. Roseblade, Subgroups of direct products of free groups, J. London Math. Soc. (2) 30 (1984), 44–52, https://doi.org/10.1112/jlms/s2-30.1.44

where it is proven that subgroups of direct products of (virtually) free groups are "almost never" finitely presented.

Let $F$ denote the group with the presentation $\langle x_1, x_2, x_3| x_1^2=x_2^2=x_3^2=1\rangle$. This group can be embedded as a discrete subgroup of $SU(1,1)$ (the 2-fold cover of $PU(1,1)$, the isometry group of the hyperbolic plane). Next, let $G=F\times F$. Then $G$ embeds discretely in $SU(1,1)\times SU(1,1)\subset SL(4, {\mathbb C})$. Consider homomorphism $\phi: F\to {\mathbb Z}$ sending $x_1, x_2$ to $0$ and $x_3$ to $1$. Let $\Phi=(\phi,\phi): G\to {\mathbb Z}$. Define $\Gamma$ to be the kernel of $\Phi$. Then $\Gamma$ is generated by the following elements: $$ (x_1,1), (1,x_1), (x_2,1), (1,x_2), (x_3, x_3^{-1}). $$ Furthermore, $\Gamma$ is not finitely presentable. See [1] where a similar example is explained in detail. Now, every generator of $\Gamma$ is an involution.

Example 2. This is a variation on the example in Michael Kapovich' On normal subgroups in the fundamental groups of complex surfaces https://arxiv.org/abs/math/9808085

Start with the complex-hyperbolic lattice $G$ in $SU(2,1)\subset SL(3, {\mathbb C})$ which appears at number 9 on Thurston's list in [3], page 38.

[3] Shapes of polyhedra and triangulations of the sphere, Geom. Topol. Monogr. 1 (1998), 511-549, https://doi.org/10.2140/gtm.1998.1.511, arXiv:math/9801088

Let $B^2$ be the complex 2-ball on which $G$ acts holomorphically. Geometry of the quotient was understood in detail by Hirzebruch et al. The quotient $B^2/G$, as an orbifold, admits a singular holomorphic fibration whose base is the projective line with three singular points of order $5$, while the generic fiber $\Sigma$ is the projective line with four singular points of order $5$. Now, let $\Gamma\subset G$ be the image of the fundamental group of the generic fiber $\Sigma$ in $G$. The group $\Gamma$ is generated by four complex reflections of order $5$ (each has two eigenvalues $1$ and one which is 5th root of unity). Then, using the main result of [2], one shows that $\Gamma$ is not finitely presentable. (The proof in [2] applies to torsion-free subgroup of finite index in $\Gamma$, but finite presentability is preserved by passing to finite index subgroups.)

So, what is the difference with the result on real reflection groups? The key is that Tits-Vinberg theorem applies to groups $\Gamma$ generated by real reflections in faces of a convex cone $\Delta$ where "angles'' (whatever this means) have the form $\pi/m_i$. The cone $\Delta$ serves as a fundamental domain for the Coxeter group $\Gamma$ and one can read off the presentation of $\Gamma$ from the combinatorics and geometry of $\Delta$. Such a cone is completely missing in the case of groups generated by complex reflections. Even in the real case, once you drop the "$\pi/m_i$" assumption, the linear representation $\psi$ of the Coxeter group $\Gamma$ (whose presentation is defined by repeating the presentation which appears in the $\pi/m_i$ case) will fail to be faithful. By working harder, one can almost surely get examples of groups $\psi(\Gamma)$ which are not finitely-presented as well.


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