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Algebraic fundamental groups of fake projective planes

Matthew Stover

Abstract

Fundamental groups of fake projective planes fall into fifty distinct isomorphism classes, one for each complex conjugate pair. We prove that this is not the case for their algebraic fundamental groups: there are only forty-six isomorphism classes. We show that there are four pairs of complex conjugate pairs of fake projective planes that are $\mathrm{Aut}(\mathbb{C})$-equivalent and hence have mutually isomorphic algebraic fundamental groups. All other pairs of algebraic fundamental groups are shown to be distinct through explicit finite étale covers. As a by-product, this provides the first examples of commensurable but nonisomorphic lattices in a rank one semisimple Lie group that have isomorphic profinite completions.

Algebraic fundamental groups of fake projective planes

Abstract

Fundamental groups of fake projective planes fall into fifty distinct isomorphism classes, one for each complex conjugate pair. We prove that this is not the case for their algebraic fundamental groups: there are only forty-six isomorphism classes. We show that there are four pairs of complex conjugate pairs of fake projective planes that are -equivalent and hence have mutually isomorphic algebraic fundamental groups. All other pairs of algebraic fundamental groups are shown to be distinct through explicit finite étale covers. As a by-product, this provides the first examples of commensurable but nonisomorphic lattices in a rank one semisimple Lie group that have isomorphic profinite completions.
Paper Structure (30 sections, 16 theorems, 53 equations, 6 figures, 10 tables)

This paper contains 30 sections, 16 theorems, 53 equations, 6 figures, 10 tables.

Table of Contents

  1. Introduction
  2. The action of $\mathop{\mathrm{Aut}}\nolimits(\mathbb{C})$
  3. Profinite completions of discrete groups
  4. Distinct isomorphism classes
  5. Conjugation of Shimura varieties
  6. Buildings and their vertices
  7. The building for $\mathop{\mathrm{SL}}\nolimits_3(F_\nu)$
  8. The building for $\mathop{\mathrm{SU}}\nolimits(h, E_\mu)$
  9. Isomorphic algebraic fundamental groups
  10. The pair $\{34, 35\}$
  11. The number fields and some prime ideals
  12. The algebraic group and open compact subgroups
  13. The arithmetic groups
  14. The isomorphism of profinite completions for $j = 34, 35$
  15. The pair $\{43, 44\}$
  16. ...and 15 more sections

Key Result

Theorem 1.1

There are exactly forty-six isomorphism classes of algebraic fundamental groups of fake projective planes. Specifically, suppose that $X$ and $Y$ are fake projective planes and that $\pi_1(X) = \Gamma_j, \pi_1(Y) = \Gamma_k$ for $j < k$ under the numbering given in the Appendix. Then $\pi_1^{\mathrm

Figures (6)

  • Figure 1: A hierarchy for identifying fake projective planes
  • Figure 2: The diagram of fake projective plane groups for $j = 32, 34, 35$
  • Figure 3: The common $\mathbb{Z} / 3$ cover of $X_{32}$, $X_{34}$, and $X_{35}$
  • Figure 4: The diagram of fake projective plane groups for $j = 42, 43, 44$
  • Figure 5: The common $\mathbb{Z} / 3$ cover of $X_{43}$ and $X_{44}$
  • ...and 1 more figures

Theorems & Definitions (29)

  • Theorem 1.1
  • Corollary 1.2
  • Corollary 1.3
  • Remark 1.6
  • Remark 1.7
  • Remark 1.8: Remark on use of the computer
  • Lemma 2.1
  • proof
  • Corollary 2.2
  • proof
  • ...and 19 more