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Recognition and constructive membership for discrete subgroups of $\mathrm{SL}_2(\mathbb{R})$

Ari Markowitz

Abstract

We provide algorithms to decide whether a finitely generated subgroup of $\mathrm{SL}_2(\mathbb{R})$ is discrete, solve the constructive membership problem for finitely generated discrete subgroups of $\mathrm{SL}_2(\mathbb{R})$, and compute a fundamental domain for a finitely generated Fuchsian group. These algorithms have been implemented in Magma for groups defined over real algebraic number fields.

Recognition and constructive membership for discrete subgroups of $\mathrm{SL}_2(\mathbb{R})$

Abstract

We provide algorithms to decide whether a finitely generated subgroup of is discrete, solve the constructive membership problem for finitely generated discrete subgroups of , and compute a fundamental domain for a finitely generated Fuchsian group. These algorithms have been implemented in Magma for groups defined over real algebraic number fields.

Paper Structure

This paper contains 19 sections, 40 theorems, 26 equations, 14 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Reduced generating sets
  5. Bounding paths
  6. Self-intersecting paths
  7. A bound on the displacement of a replacement
  8. The recognition algorithm
  9. Performing exact computations
  10. Properties of reduced sets
  11. Bounding polygons
  12. The Nielsen region
  13. The signature
  14. The constructive membership problem
  15. Discrete subgroups of $\mathop{\mathrm{SL}}\nolimits_2(\mathbb{R})$ and $\mathop{\mathrm{PSL}}\nolimits_2(\mathbb{R})$
  16. ...and 4 more sections

Key Result

Theorem A

Let $X$ be a finite subset of $\mathop{\mathrm{SL}}\nolimits_2(\mathbb{R})$ or $\mathop{\mathrm{PSL}}\nolimits_2(\mathbb{R})$. There exists an algorithm that decides whether or not $X$ generates a discrete group.

Figures (14)

  • Figure 1: The embedding of a Cayley graph in $\mathbb{H}^2$
  • Figure 2: An infinite leftmost path $\mathop{\mathrm{\mathcal{L}}}\nolimits(e)$ in \ref{['fig:cayley-graph']}
  • Figure 3: The embedding of a worse-chosen Cayley graph of the group in \ref{['fig:cayley-graph']}
  • Figure 4: The construction of $P$ for \ref{['prop:path-impl-clock']}
  • Figure 5: The construction in \ref{['lem:intersection0']}
  • ...and 9 more figures

Theorems & Definitions (82)

  • Theorem A
  • Theorem B
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Proposition 3.4
  • proof
  • Definition 3.5
  • Definition 3.6
  • Definition 3.7
  • ...and 72 more