One-dimensional actions of Higman’s group

One-dimensional actions of Higman’s group, Discrete Analysis 2019:20, 15 pp.

In 1951 Higman constructed the first known example of an infinite finitely generated simple group. He began with the group $H$ that has presentation $\langle a,b,c,d|aba^{-1}=b^2,bcb^{-1}=c^2$, $cdc^{-1}=d^2,dad^{-1}=a^2\rangle$. A theorem of Neumann states that any finitely generated group contains a maximal proper normal subgroup. The quotient of $H$ by such a subgroup must be simple, and Higman proved that any finite quotient of $H$ must be trivial. Therefore, $H$ has a quotient that is both simple and infinite, and this quotient is clearly finitely generated.

Later, Thompson went further, constructing an infinite simple group (traditionally known as $T$) of homeomorphisms of the circle that is not only finitely generated but even finitely presented. (Note that a quotient of a finitely presented group need not be finitely presented, so Higman’s group cannot, as far as anyone knows, be used for this purpose.) Since then, several other interesting infinite simple groups of homeomorphisms of the real line or the circle have been discovered.

A theorem of Malcev states that a finitely generated subgroup of GL$_n(K)$, where $K$ is any field, has the property that for every pair of elements $x,y$ there is a finite quotient in which $x$ and $y$ are not identified, which is equivalent to saying that every non-identity element has a non-trivial image in some quotient. (Such groups are called residually finite). In particular, a non-trivial finitely generated subgroup of GL$_n(K)$ has a non-trivial finite quotient. It follows that an infinite simple group cannot have any non-trivial finite-dimensional linear representations.

In this situation, it is natural to wonder whether there are other ‘nice’ actions of $H$, and that is the question addressed in this paper. Given that several simple infinite groups arise as groups of homeomorphisms of the real line or the circle, it is natural to look for actions of that kind, which is exactly what the authors of this paper do: they find a faithful action of $H$ on the real line by homeomorphisms. That is, they find four homeomorphisms $a,b,c,d$ of $\mathbb R$ that satisfy the defining relations of $H$ and satisfy no relations that are not satisfied in $H$. This is in a sense the reverse of the usual process: they start with a presentation and find an action by homeomorphisms, rather than starting with a group of homeomorphisms and finding a presentation. In the negative direction, they prove that they cannot replace the homeomorphisms in their result by $C^1$ diffeomorphisms, either on the line or on the circle.

It turns out that a countable group has a faithful action by homeomorphisms on the real line if and only if it is left-orderable, which means that there is a total order on the group such that if $x<y$ then $zx<zy$ for every $z$. Other groups that have been shown to be left-orderable are the braid groups (a result of Dehornoy) and the right-angled Artin groups (an exercise). In those two cases, the groups turned out to be linear, but that was proved well after they were known to be left-orderable.

The result of this paper turns out to have interesting further consequences. One is that if $R$ is any ring with no zero divisors, then so is the group ring $R[H]$, which shows that $H$ satisfies a conjecture of Kaplansky. And a by-product of the proof is the construction of several $\mathbb Z$-valued unbounded quasimorphisms, which are maps $\phi:H\to\mathbb R$ such that there exists $D$ for which $|\phi(x)+\phi(y)-\phi(xy)|\leq D$ for every $x,y\in H$. (It follows from abstract considerations that the quotient of the space of $\mathbb R$-valued quasimorphisms by the space of bounded functions is of continuum-sized dimension, but that does not imply that any of them are $\mathbb Z$-valued.)