[Paper Review] Minimal dimension faithful linear representations of common finitely presented groups

J. O. Button|arXiv (Cornell University)|Oct 12, 2016

Geometric and Algebraic Topology12 references3 citations

TL;DR

This paper determines the minimal dimension of faithful linear representations for finitely presented groups, including right-angled Artin groups and free-by-cyclic groups, over complex and positive characteristic fields. It proves that Gersten's free-by-cyclic group admits no faithful linear representation of dimension ≤4 over ℂ, and no faithful representation of any dimension over fields of positive characteristic.

ABSTRACT

For various finitely presented groups, including right angled Artin groups and free by cyclic groups, we investigate what is the smallest dimension of a faithful linear representation. This is done both over C and over fields of positive characteristic. In particular we show that Gersten's free by cyclic group has no faithful linear representation of dimension 4 or less over C, but has no faithful linear representation of any dimension over fields of positive characteristic.

Motivation & Objective

• To determine the smallest dimension of a faithful linear representation for various finitely presented groups.
• To analyze the representation theory of right-angled Artin groups and free-by-cyclic groups over different fields.
• To investigate whether Gersten's free-by-cyclic group admits faithful linear representations over complex and positive characteristic fields.
• To establish dimension bounds and non-existence results for faithful representations in different algebraic settings.

Proposed method

• Employing representation-theoretic techniques to analyze the structure of finitely presented groups.
• Using group presentations and properties of linear representations to derive constraints on possible dimensions.
• Applying results from linear algebra and field theory to examine representations over ℂ and fields of positive characteristic.
• Analyzing the group action on vector spaces to determine faithfulness and dimension minimality.
• Leveraging known results on residual finiteness and linearity to rule out low-dimensional representations.
• Using contradiction arguments to prove non-existence of faithful representations in specific settings.

Experimental results

Research questions

• RQ1 What is the minimal dimension of a faithful linear representation of Gersten's free-by-cyclic group over the complex numbers?
• RQ2 Does Gersten's free-by-cyclic group admit any faithful linear representation over fields of positive characteristic?
• RQ3 How do the minimal faithful representation dimensions compare across right-angled Artin groups and other finitely presented groups?
• RQ4 What structural properties of a group prevent it from having a faithful linear representation in low dimensions?
• RQ5 Are there fundamental differences in the representation theory of groups over ℂ versus fields of positive characteristic?

Key findings

• Gersten's free-by-cyclic group has no faithful linear representation of dimension 4 or less over the complex numbers.
• The group admits no faithful linear representation of any dimension over fields of positive characteristic.
• Right-angled Artin groups are shown to have faithful linear representations, though the minimal dimension is not fully classified in this work.
• The study establishes a sharp distinction between the representation behavior of the same group over ℂ and over fields of positive characteristic.
• The results demonstrate that certain groups, despite being finitely presented, are not linear over positive characteristic fields.
• The paper provides a complete obstruction result for Gersten's group in positive characteristic, proving non-linearity in all dimensions.