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Residual finiteness and discrete subgroups of Lie groups

Matthew Stover

TL;DR

This survey investigates when lattices Γ in connected real Lie groups G are residually finite, contrasting linear/solvable cases (where RF often holds) with the semisimple setting where the situation is intricate. It synthesizes higher-rank phenomena driven by arithmeticity, CSP, and the metaplectic kernel, showing many lattices fail RF in large central covers, while rank-one cases exhibit a spectrum of results, including complete RF in covers of PSL$_2(oldsymbol{R})$ and Sp$(n,1)$ and substantial positive results for PU$(n,1)$ in several arithmetic scenarios. It also outlines major open problems for nonarithmetic and division-algebra lattices in PU$(n,1)$ and the elusive case of $ ext{F}_4^{(-20)}$, and it connects RF to broader questions in geometric group theory via Gromov hyperbolicity and Dehn-filling techniques. Finally, it reduces the general RF question to the semisimple component through Levi decomposition, emphasizing that the semisimple part largely controls whether lattices are residually finite.

Abstract

Let $G$ be a real Lie group and $Γ< G$ be a discrete subgroup of $G$. Is $Γ$ residually finite? This paper describes known positive and negative results then poses some questions whose answers will lead to a fairly complete answer for lattices.

Residual finiteness and discrete subgroups of Lie groups

TL;DR

This survey investigates when lattices Γ in connected real Lie groups G are residually finite, contrasting linear/solvable cases (where RF often holds) with the semisimple setting where the situation is intricate. It synthesizes higher-rank phenomena driven by arithmeticity, CSP, and the metaplectic kernel, showing many lattices fail RF in large central covers, while rank-one cases exhibit a spectrum of results, including complete RF in covers of PSL and Sp and substantial positive results for PU in several arithmetic scenarios. It also outlines major open problems for nonarithmetic and division-algebra lattices in PU and the elusive case of , and it connects RF to broader questions in geometric group theory via Gromov hyperbolicity and Dehn-filling techniques. Finally, it reduces the general RF question to the semisimple component through Levi decomposition, emphasizing that the semisimple part largely controls whether lattices are residually finite.

Abstract

Let be a real Lie group and be a discrete subgroup of . Is residually finite? This paper describes known positive and negative results then poses some questions whose answers will lead to a fairly complete answer for lattices.
Paper Structure (13 sections, 30 theorems, 26 equations, 3 tables)

This paper contains 13 sections, 30 theorems, 26 equations, 3 tables.

Table of Contents

  1. Definitions and first examples
  2. Basic facts
  3. Residual finiteness
  4. Lie groups
  5. Semisimple groups
  6. Higher rank
  7. Rank one
  8. Full understanding: $\mathop{\mathrm{SO}}\nolimits^\circ(n,1)$ and $\mathop{\mathrm{Sp}}\nolimits(n,1)$
  9. Positive results for $\mathop{\mathrm{PU}}\nolimits(n,1)$
  10. Remaining cases for $\mathop{\mathrm{PU}}\nolimits(n,1)$
  11. The wide open case of $\mathrm{F}_4^{(-20)}$
  12. Gromov hyperbolic groups
  13. The general case

Key Result

Theorem 1.2

Let $\Gamma$ be a finitely generated subgroup of $\mathop{\mathrm{GL}}\nolimits_N(\mathbb{C})$ for some $N \ge 1$. Then $\Gamma$ is residually finite.

Theorems & Definitions (47)

  • Theorem 1.2: Malcev MalcevRF
  • Example 1.3: Folklore
  • Example 1.4: Cremaschi and Souto CremaschiSouto
  • Theorem 1.5: Deligne DeligneSp
  • Theorem 1.8
  • proof
  • Remark 1.9
  • Lemma 2.1
  • Corollary 2.2
  • Lemma 2.3: Malcev MalcevSemi
  • ...and 37 more