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2-chains and square roots of Thompson’s group $F$

Part of: Special aspects of infinite or finite groups Low-dimensional topology PL-topology Low-dimensional dynamical systems

Published online by Cambridge University Press:  25 March 2019

THOMAS KOBERDA Open the ORCID record for THOMAS KOBERDA [Opens in a new window]  and
YASH LODHA
THOMAS KOBERDA
Affiliation:
Department of Mathematics, University of Virginia, Charlottesville, VA22904-4137, USA email [email protected]
YASH LODHA
Affiliation:
EPFL SB MATH EGG, Station 8, MA C3 584 (Bâtiment MA), Station 8, CH-1015, Lausanne, Switzerland email [email protected]
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Abstract

We study 2-generated subgroups $\langle f,g\rangle <\operatorname{Homeo}^{+}(I)$ such that $\langle f^{2},g^{2}\rangle$ is isomorphic to Thompson’s group $F$, and such that the supports of $f$ and $g$ form a chain of two intervals. We show that this class contains uncountably many isomorphism types. These include examples with non-abelian free subgroups, examples which do not admit faithful actions by $C^{2}$ diffeomorphisms on 1-manifolds, examples which do not admit faithful actions by $PL$ homeomorphisms on an interval, and examples which are not finitely presented. We thus answer questions due to Brin. We also show that many relatively uncomplicated groups of homeomorphisms can have very complicated square roots, thus establishing the behavior of square roots of $F$ as part of a general phenomenon among subgroups of $\operatorname{Homeo}^{+}(I)$.

Keywords

Thompson’s grouporderable groupequations over homeomorphism groupssmoothable group actions

MSC classification

Primary: 57M60: Group actions in low dimensions
Secondary: 37E05: Maps of the interval (piecewise continuous, continuous, smooth) 57Q99: None of the above, but in this section 57S25: Groups acting on specific manifolds 20F60: Ordered groups 20F14: Derived series, central series, and generalizations
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 40 , Issue 9 , September 2020 , pp. 2515 - 2532
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Copyright
© Cambridge University Press, 2019

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