Simple left-orderable groups with prescribed finiteness length

For every integer n ≥ 2 we construct an infinite simple left-orderable group of type F_n and not of type FP_{n+1}(ℤ), and hence not of type F_{n+1}, where type F_n means that a classifying space has finite n-skeleton. This answers affirmatively the left-orderable interpretation of a question of Xiaolei Wu in the problem list associated with the 2024 CIRM meeting. The construction inserts arithmetic groups of one-sided germs into a group of homeomorphisms of the line. A forest complex controls the finiteness of the germ groups, and an actual interval-lift argument controls the singularity stabilizers. An (n−1)-connected band in the germ complex then gives type F_n for the simple derived subgroup. A Lipschitz retraction onto a kernel of endpoint characters retains the homological finiteness obstruction.

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