Let $G= \def\<#1>{\left<#1\right>}\<a,b,c,d \mid \text{finite number of defining relations}>$ be a left-orderable group. Let $H=\<a,b \mid \text{a partial subset of defining relations of $G$}>$ a convex subgroup of $G$ relative to some left-order $<$. Then $G$ acts by orientation preserving homeomorphisms on the real line (as it is countable).
Is it necessarily true that:
- $H$ admits a fixed point $x \in \mathbf R$
- the subgroup generated by $c$, $d$ (with another partial subset of defining relations of $G$) acts freely on this $x$?
I would appreciate very much if you could also refer me to the relevant literature.
In general, the answer is negative, but you are most probably referring to the action produced by the order (the so-called dynamical realization). In that case, the answer for 1. is YES. For 2. it is not true: consider for instance a lexicographic order on Z^4. What is true is that the action on this fixed point has neither lower nor upper limit. More information is in http://arxiv.org/abs/1408.5805