Volume 18, issue 4 (2018)

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Quasiautomorphism groups of type $F_\infty$

Samuel Audino, Delaney R Aydel and Daniel S Farley

Algebraic & Geometric Topology 18 (2018) 2339–2369
Abstract

The groups $QF\phantom{\rule{0.3em}{0ex}}$, $QT\phantom{\rule{0.3em}{0ex}}$, $\stackrel{̄}{Q}T\phantom{\rule{0.3em}{0ex}}$, $\stackrel{̄}{Q}V$ and $QV$ are groups of quasiautomorphisms of the infinite binary tree. Their names indicate a similarity with Thompson’s well-known groups $F\phantom{\rule{0.3em}{0ex}}$, $T$ and $V\phantom{\rule{0.3em}{0ex}}$.

We will use the theory of diagram groups over semigroup presentations to prove that all of the above groups (and several generalizations) have type ${F}_{\infty }$. Our proof uses certain types of hybrid diagrams, which have properties in common with both planar diagrams and braided diagrams. The diagram groups defined by hybrid diagrams also act properly and isometrically on $CAT\left(0\right)$ cubical complexes.

Keywords
quasiautomorphism group, Thompson's groups, Houghton groups, finiteness properties of groups, CAT(0) cubical complexes
Primary: 20F65
Secondary: 57M07