Solvable lattices and their groups of automorphisms

G. A. Soifer

doi:10.1081/agb-120027951

Solvable lattices and their groups of automorphisms

G. A. Soifer

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

2 Scopus citations

Abstract

The main goal of this paper is to construct an "algebraic" representation of a group automorphisms Aut Γ for any elementary solvable group Γ. "Algebraic" means that the image of a semisimple (unipotent) automorphism, in the sense of Wang, will be a semisimple (unipotent) matrix Theorem 1. This gives an answer on the question asking by Wang. As a corollary of this theorem, we show that for any group of automorphisms Aut Γ of a lattice Γ in a solvable connected group Lie G there exist a representation ρ : Aut Γ → GL_n(ℤ), such that ρ(Aut Γ) is an arithmetic subgroup in the Zariski closure ρ(Aut Γ).

Original language	English
Pages (from-to)	805-817
Number of pages	13
Journal	Communications in Algebra
Volume	32
Issue number	3
DOIs	https://doi.org/10.1081/agb-120027951
State	Published - 2004

Bibliographical note

Funding Information:
It is also a pleasure for me to thank the Isaac Newton Mathematical Institute, University of Cambridge and SFB343, Bielefeld University for their support and hospitality, the Emmy Noether Research Institute for Mathematics, Bar-Ilan University, and the Center of Excellence, supported by ISF, grant N 8008/02-1.

Funding

It is also a pleasure for me to thank the Isaac Newton Mathematical Institute, University of Cambridge and SFB343, Bielefeld University for their support and hospitality, the Emmy Noether Research Institute for Mathematics, Bar-Ilan University, and the Center of Excellence, supported by ISF, grant N 8008/02-1.

Funders	Funder number
Emmy Noether Research Institute for Mathematics, Bar-Ilan University
University of Cambridge	SFB343
Israel Science Foundation	N 8008/02-1
Universität Bielefeld

Keywords

Arithmetic subgroup
Automorphism
Lattice

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10.1081/agb-120027951

Cite this

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note = "Funding Information: It is also a pleasure for me to thank the Isaac Newton Mathematical Institute, University of Cambridge and SFB343, Bielefeld University for their support and hospitality, the Emmy Noether Research Institute for Mathematics, Bar-Ilan University, and the Center of Excellence, supported by ISF, grant N 8008/02-1.",

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AB - The main goal of this paper is to construct an "algebraic" representation of a group automorphisms Aut Γ for any elementary solvable group Γ. "Algebraic" means that the image of a semisimple (unipotent) automorphism, in the sense of Wang, will be a semisimple (unipotent) matrix Theorem 1. This gives an answer on the question asking by Wang. As a corollary of this theorem, we show that for any group of automorphisms Aut Γ of a lattice Γ in a solvable connected group Lie G there exist a representation ρ : Aut Γ → GLn(ℤ), such that ρ(Aut Γ) is an arithmetic subgroup in the Zariski closure ρ(Aut Γ).

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KW - Automorphism

KW - Lattice

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Solvable lattices and their groups of automorphisms

Abstract

Bibliographical note

Funding

Keywords

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