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Solvability of the ${\rm SK}_1$-analog of the orthogonal groups

Ambily A. A., Gayathry Pradeep

TL;DR

This work extends the framework of local-global and dilation principles from classical and symplectic/orthogonal groups to the odd orthogonal setting, by developing the relative DSER elementary orthogonal group ${\rm EO}_{(R,I)}(Q,\mathbb{H}(P))$ and proving its normality in the full orthogonal group. A key technical tool is the relative dilation principle, which, via excision rings and local-global arguments, yields a relative local-global principle and enables explicit solvability and nilpotency results for the ${\rm SK}_1$-analogs of these groups in the odd dimension. For local rings, the quotient ${\rm SO}_{2m+1}(R[X])/{\rm EO}_{2m+1}(R[X])$ is shown to be solvable of length at most $2$, and the corresponding relative SK$_1$-quotients have similar bounds with nilpotency class at most $2$. The paper also establishes a commutativity principle for odd elementary orthogonal groups in the homotopy setting and leverages equality results ${\rm EO}_R(Q,\mathbb{H}(P)) = {\rm EO}_{n+2m}(R)$ in the free case to connect DSER theory with classical elementary groups. Overall, the results deepen the understanding of the structure of SK$_1$-analogs for orthogonal groups and extend foundational principles to the odd-dimensional context.

Abstract

We prove the dilation principle for the relative Dickson-Siegel-Eichler-Roy (DSER) elementary orthogonal group and using the dilation principle we prove the Quillen's analog of the local-global principle for the group. Applying the relative local-global principle, we prove the solvability and nilpotency of the ${\rm SK_1}$-analog of the orthogonal groups and study the homotopy and commutativity principle for odd elementary orthogonal groups.

Solvability of the ${\rm SK}_1$-analog of the orthogonal groups

TL;DR

This work extends the framework of local-global and dilation principles from classical and symplectic/orthogonal groups to the odd orthogonal setting, by developing the relative DSER elementary orthogonal group and proving its normality in the full orthogonal group. A key technical tool is the relative dilation principle, which, via excision rings and local-global arguments, yields a relative local-global principle and enables explicit solvability and nilpotency results for the -analogs of these groups in the odd dimension. For local rings, the quotient is shown to be solvable of length at most , and the corresponding relative SK-quotients have similar bounds with nilpotency class at most . The paper also establishes a commutativity principle for odd elementary orthogonal groups in the homotopy setting and leverages equality results in the free case to connect DSER theory with classical elementary groups. Overall, the results deepen the understanding of the structure of SK-analogs for orthogonal groups and extend foundational principles to the odd-dimensional context.

Abstract

We prove the dilation principle for the relative Dickson-Siegel-Eichler-Roy (DSER) elementary orthogonal group and using the dilation principle we prove the Quillen's analog of the local-global principle for the group. Applying the relative local-global principle, we prove the solvability and nilpotency of the -analog of the orthogonal groups and study the homotopy and commutativity principle for odd elementary orthogonal groups.

Paper Structure

This paper contains 7 sections, 35 theorems, 40 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Normality of relative DSER elementary orthogonal group
  4. Relative Dilation Principle and local-global principle
  5. Equality of DSER elementary orthogonal group and usual elementary orthogonal group in free case
  6. ${\rm SK_1}$-analog of the relative DSER elementary orthogonal group
  7. Commutativity principle for odd elementary orthogonal group

Key Result

Theorem 1.1

Let $R$ be a commutative ring with finite Bass-Serre dimension $\delta(R)$, and $S$ be an associative quasi-finite $R$-algebra. Then the group ${\rm K}_{1,n}(S)$ with $n\geq 3$ is nilpotent by abelian and the nilpotency class is at most $\delta(R)+2-n$.

Theorems & Definitions (68)

  • Theorem 1.1: Bak1991
  • Theorem 1.2: RRaoSS2017
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5: RRaoSS2022
  • Theorem 1.6
  • Definition 2.1
  • Definition 2.2: Relative orthogonal group
  • Definition 2.3: DSER elementary orthogonal transformations
  • Definition 2.4: Elementary orthogonal group
  • ...and 58 more