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On the homology of special unitary groups over polynomial rings

Claudio Bravo

TL;DR

This work extends homotopy-invariance results from isotrivial groups to the smallest non-isotrivial quasi-split SU_3 group scheme over \\mathbb{P}^1_F. It proves a natural injection \\iota: \\mathrm{PGL}_2(F) \\\hookrightarrow \\mathrm{SU}_3(F[t]) that induces an isomorphism on integral homology, and it provides a detailed Mayer-Vietoris/Bass-Serre analysis to describe the rational homology of \\mathrm{SU}_3(F[t,t^{-1}]). Central to the approach are explicit fundamental domains for the action on Bruhat-Tits buildings and the resulting amalgamated-product decompositions, enabling reductions to simpler groups such as \\mathrm{PGL}_2(F) and \\mathrm{SL}_2(F). The results generalize prior work of Knudson and Wendt to a non-isotrivial setting and lay groundwork for further extensions to non-isotrivial quasi-split group schemes over function fields.

Abstract

In this work, we answer the homotopy invariance question for the ''smallest'' non-isotrivial group-scheme over $\mathbb{P}^1$, obtaining a result, which is not contained in previous works due to Knudson and Wendt. More explicitly, let $\mathcal{G}=\mathrm{SU}_{3,\mathbb{P}^1}$ be the (non-isotrivial) non-split group-scheme over $\mathbb{P}^1$ defined from the standard (isotropic) hermitian form in three variables. In this article, we prove that there exists a natural homomorphism $\mathrm{PGL}_2(F) \to \mathcal{G}(F[t])$ that induces isomorphisms $H_*(\mathrm{PGL}_2(F), \mathbb{Z}) \to H_*(\mathcal{G}(F[t]), \mathbb{Z})$. Then we study the rational homology of $\mathcal{G}(F[t,t^{-1}])$, by previously describing suitable fundamental domains for certain arithmetic subgroups of $\mathcal{G}$.

On the homology of special unitary groups over polynomial rings

TL;DR

This work extends homotopy-invariance results from isotrivial groups to the smallest non-isotrivial quasi-split SU_3 group scheme over \\mathbb{P}^1_F. It proves a natural injection \\iota: \\mathrm{PGL}_2(F) \\\hookrightarrow \\mathrm{SU}_3(F[t]) that induces an isomorphism on integral homology, and it provides a detailed Mayer-Vietoris/Bass-Serre analysis to describe the rational homology of \\mathrm{SU}_3(F[t,t^{-1}]). Central to the approach are explicit fundamental domains for the action on Bruhat-Tits buildings and the resulting amalgamated-product decompositions, enabling reductions to simpler groups such as \\mathrm{PGL}_2(F) and \\mathrm{SL}_2(F). The results generalize prior work of Knudson and Wendt to a non-isotrivial setting and lay groundwork for further extensions to non-isotrivial quasi-split group schemes over function fields.

Abstract

In this work, we answer the homotopy invariance question for the ''smallest'' non-isotrivial group-scheme over , obtaining a result, which is not contained in previous works due to Knudson and Wendt. More explicitly, let be the (non-isotrivial) non-split group-scheme over defined from the standard (isotropic) hermitian form in three variables. In this article, we prove that there exists a natural homomorphism that induces isomorphisms . Then we study the rational homology of , by previously describing suitable fundamental domains for certain arithmetic subgroups of .

Paper Structure

This paper contains 12 sections, 37 theorems, 82 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main results
  3. Conventions and preliminaries
  4. The radical datum of SU3
  5. The Bruhat-Tits tree of SU3
  6. On the action of G on the Bruhat-Tits tree
  7. The homology of SU31
  8. SU32
  9. On the action of G on the Bruhat-Tits tres
  10. The homology of the Hecke congruence subgroup G0
  11. Fundamental domain for some congruence subgroups of G
  12. On the homology

Key Result

Theorem 1.1

Knudson1 Let $F$ be a field with $\mathrm{char}(F)=0$. The canonical inclusion $\mathrm{SL}_2(F) \hookrightarrow \mathrm{SL}_2(F[t])$ induces isomorphisms $H_*(\mathrm{SL}_2(F),\mathbb{Z}) \xrightarrow{\cong} H_*(\mathrm{SL}_2(F[t]),\mathbb{Z})$.

Figures (1)

  • Figure 1: In the left side, the fundamental chamber of the building of $\mathrm{SL}_3$. In the right side, the tree of $\mathrm{SU}_3$ insides the building for $\mathrm{SL}_3$.

Theorems & Definitions (63)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Lemma 3.1
  • Remark 3.2
  • Lemma 3.3
  • ...and 53 more