Ascending chains of irreducible lattices, bi-reversible automata and affine arithmetic groups

2026/07/16 by Pierre-Emmanuel Caprace, Justin Vast
#math.GR

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Abstract

For each n ≥ 2, we construct an ascending chain of irreducible lattices in the product of n homogeneous trees. Moreover, for each pair of integers m₁, m₂ ≥ 1, we define explicitly a bi-reversible automaton \mathcal B such that the group G_{\mathcal B} defined by the automaton \mathcal B has finiteness length m₁ (i.e. it is of type F_m1 but not of type FP_m1+1), and the group G_{\mathcal B^*} defined by the dual automaton has finiteness length m₂. Both constructions rely on the consideration of S-arithmetic groups in the affine group of a global function field.

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