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Thinness of some hypergeometric groups in Sp(6)

Sandip Singh, Shashank Vikram Singh

Abstract

We show that the hypergeometric groups corresponding to the seven pairs of the parameters $α$, $β$ where $α$ = (0, 0, 0, 0, 0, 0) and $β$ is any of the parameters (1/2, 1/2, 1/2, 1/2, 1/2, 1/2), (1/2, 1/2, 1/2, 1/2, 1/3, 2/3), (1/2, 1/2, 1/2, 1/2, 1/4, 3/4), (1/2, 1/2, 1/2, 1/2, 1/6, 5/6), (1/2, 1/2, 1/3, 2/3, 1/3, 2/3), (1/2, 1/2, 1/3, 2/3, 1/4, 3/4), (1/2, 1/2, 1/5, 2/5, 3/5, 4/5) are thin.

Thinness of some hypergeometric groups in Sp(6)

Abstract

We show that the hypergeometric groups corresponding to the seven pairs of the parameters , where = (0, 0, 0, 0, 0, 0) and is any of the parameters (1/2, 1/2, 1/2, 1/2, 1/2, 1/2), (1/2, 1/2, 1/2, 1/2, 1/3, 2/3), (1/2, 1/2, 1/2, 1/2, 1/4, 3/4), (1/2, 1/2, 1/2, 1/2, 1/6, 5/6), (1/2, 1/2, 1/3, 2/3, 1/3, 2/3), (1/2, 1/2, 1/3, 2/3, 1/4, 3/4), (1/2, 1/2, 1/5, 2/5, 3/5, 4/5) are thin.
Paper Structure (10 sections, 2 theorems, 43 equations, 1 table)

This paper contains 10 sections, 2 theorems, 43 equations, 1 table.

Table of Contents

  1. Introduction
  2. Notation and Strategy
  3. Proof of the freeness of the hypergeometric groups of Table \ref{['Table-1']}
  4. Case 1
  5. Case 2
  6. Case 3
  7. Case 4
  8. Case 5
  9. Case 6
  10. Case 7

Key Result

Theorem 1.2

The hypergeometric groups $\Gamma(\alpha,\beta)$ associated to the $7$ pairs of the parameters $\alpha,\beta$ where $\alpha=(0,0,0,0,0,0)$ and $\beta$ is any of the $7$ parameters appearing in Table Table-1 are thin.

Theorems & Definitions (6)

  • Remark 1.1
  • Theorem 1.2
  • Theorem 1.3: Ping-Pong Lemma
  • Remark 1.4
  • Remark 2.1
  • Definition 2.2