A geometric proof that SL₂(ℤ[t,t⁻¹]) is not finitely presented

Abstract

We give a new proof of the theorem of Krstić-McCool from the title. Our proof has potential applications to the study of finiteness properties of other subgroups of SL 2 resulting from rings of functions on curves.20F05; 20F65

Homology and finiteness properties of SL2(ℤ[t,t−1])

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We show that the group H2(SL2(Z[t, t −1 ]); Z) is not finitely generated, answering a question mentioned by Bux and Wortman in [2].

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“…In [2], Bux and Wortman give a geometric proof that the group SL 2 (Z[t, t −1 ]) is not finitely presented (nor is it of type FP 2 ). They achieve this by letting the group act on a product of Bruhat-Tits trees.…”

Infinite-dimensional cohomology of SL2(Z[t,

“…We begin by recalling the structure of a Euclidean building on which Γ acts. The following construction uses the notation of Bux-Wortman in [BW06]. Let ν ∞ and ν 0 be the valuations on F (t) giving multiplicity of zeros at infinity and at zero, respectively.…”

“…The methods in this paper will be geometric. We will define two spaces on which SL 2 (J[t, t −1 ]) acts: one a Euclidean building as in [BW06], and the other a classifying space for SL 2 (J[t, t −1 ]). A map between these spaces will allow us to explicitly define an infinite family of independent cocycles in H 2 (SL 2 (J[t, t −1 ]); F ).…”

“…In [BW06], Bux-Wortman use geometric methods to prove that SL 2 (Z[t, t −1 ]) is also not F P 2 . In particular, they use the action of SL 2 (Z[t, t −1 ]) on a product of locally infinite trees.…”