Quantifying Discontinuity of Residual Group Properties Using the Weihrauch Lattice

Overview

The study investigates the properties of the map $\mathrm{Res}_{\mathcal{C}}$, which assigns to a marked group its greatest residually-$\mathcal{C}$ quotient within the space of marked groups. This map is acknowledged to be discontinuous, except in trivial cases. The research utilizes the Weihrauch lattice as a framework to quantify this observed discontinuity.

Research Context

The operation of associating a marked group with its greatest residually-$\mathcal{C}$ quotient, denoted as $\mathrm{Res}_{\mathcal{C}}$, forms a central part of this investigation. The mathematical structure within which this map operates is the space of marked groups. A key characteristic of $\mathrm{Res}_{\mathcal{C}}$ is its general discontinuity. The Weihrauch lattice serves as a metric for evaluating and categorizing this level of discontinuity, providing a formal means to quantify how discontinuous such a map can be.

The framework of 'residual properties of finitely generated groups' implies an exploration of group-theoretic structures where a group's properties are inferred from its quotients. The specific focus on the map $\mathrm{Res}_{\mathcal{C}}$ for different sets $\mathcal{C}$ indicates a comparative analysis across various classes of groups.

Approach

The research's primary approach involves applying the Weihrauch lattice to analyze the map $\mathrm{Res}_{\mathcal{C}}$. This application aims to quantify the degree of discontinuity inherent in $\mathrm{Res}_{\mathcal{C}}$.

A specific methodological aspect involves determining if 'equational noetherianity of $\mathcal{C} and 'whether the set of residually-$\mathcal{C}$ groups is a quasivariety' can be characterized by the position of $\mathrm{Res}_{\mathcal{C}}$ within the Weihrauch lattice. This suggests a correlational or definitional link being sought between lattice position and these group-theoretic properties.

The study provides 'exact classifications' of $\mathrm{Res}_{\mathcal{C}}$ for several distinct sets $\mathcal{C}$, which include:

• The set of finite groups.
• The set of nilpotent groups.
• The set of $k$-nilpotent groups, for $k\ge1$.
• The set of finitely presentable groups.
• The set of LEF groups.
• The set of torsion-free groups.

This suggests a systematic analysis of $\mathrm{Res}_{\mathcal{C}} s Weihrauch degree across a range of well-defined group classes.

Findings

The map $\mathrm{Res}_{\mathcal{C}}$, which associates a marked group with its greatest residually-$\mathcal{C}$ quotient, is found to be discontinuous, excluding trivial scenarios.

The research establishes that two specific properties—the equational noetherianity of $\mathcal{C}$ and whether the set of residually-$\mathcal{C}$ groups constitutes a quasivariety—can be characterized by the placement of $\mathrm{Res}_{\mathcal{C}}$ within the Weihrauch lattice. This indicates a direct relationship between the discontinuity quantification via the Weihrauch lattice and these inherent group-theoretic characteristics.

The study yielded exact classifications for $\mathrm{Res}_{\mathcal{C}}$ when $\mathcal{C}$ is specified as:

• The set of finite groups.
• The set of nilpotent groups.
• The set of $k$-nilpotent groups (for $k\ge1$).
• The set of finitely presentable groups.
• The set of LEF groups.
• The set of torsion-free groups.

These classifications provide concrete placements for the $\mathrm{Res}_{\mathcal{C}}$ map within the Weihrauch lattice for each of the listed group types.