[Paper Review] The Frattini subgroup for subgroups of hyperbolic groups

Ilya Kapovich|ArXiv.org|Dec 31, 2001

Geometric and Algebraic Topology15 references3 citations

TL;DR

This paper proves that for any finitely generated subgroup H of a word-hyperbolic group G, the Frattini subgroup F(H) is finite. The proof uses geometric group theory techniques, including hyperbolic geometry, quasiconvexity, and Delzant's lemma, to show that no element of infinite order in H can be a non-generator, thereby establishing F(H) is a torsion group and hence finite. This result implies that the rank and generation problems in hyperbolic groups cannot be simplified via Frattini subgroups as in nilpotent groups.

ABSTRACT

We prove that for a finitely generated subgroup $H$ of a word-hyperbolic group $G$ the Frattini subgroup $F(H)$ of $H$ is finite.

Motivation & Objective

• To establish that the Frattini subgroup F(H) of any finitely generated subgroup H of a word-hyperbolic group G is finite.
• To demonstrate that in hyperbolic groups, the Frattini subgroup does not provide simplification for the rank or generation problems, unlike in nilpotent groups.
• To analyze the structure of maximal subgroups in hyperbolic groups, particularly those not arising from finite quotients.
• To show that elements of infinite order in H cannot be non-generators, implying F(H) is a torsion group.

Proposed method

• Use the characterization that F(H) consists of all non-generators of H, i.e., elements g such that H = ⟨S⟩ implies H = ⟨S ∖ {g}⟩.
• Fix a finite generating set A for G and consider the Cayley graph X = Γ(G, A) with word metric d, assuming X is δ-hyperbolic.
• Apply Delzant's lemma to control the geometry of products of elements in H, particularly those involving powers of an infinite-order element g.
• Use quasiconvexity of ⟨g⟩ in G to establish linear growth of |g^n|_A ≥ C|n| for some C > 0.
• Construct a generating set Q = {g, h₁, ..., hₜ} for H such that Q ∖ {g} generates a proper subgroup of H, by ensuring all nontrivial products in h_j have length ≥ T > T₀.
• Establish a lower bound on the word length of any nontrivial product in the h_j's using hyperbolic geometry and the triangle inequality in the Cayley graph.

Experimental results

Research questions

• RQ1 Can the Frattini subgroup of a finitely generated subgroup of a word-hyperbolic group be infinite?
• RQ2 Do elements of infinite order in such subgroups qualify as non-generators?
• RQ3 To what extent can the Frattini subgroup be used to reduce the rank or generation problem in hyperbolic groups?
• RQ4 Are maximal subgroups in hyperbolic groups primarily pullbacks from finite quotients, or do they have a more pathological nature?

Key findings

• The Frattini subgroup F(H) of any finitely generated subgroup H of a word-hyperbolic group G is finite.
• No element of infinite order in H can be a non-generator, so F(H) is a torsion group.
• Since all torsion subgroups in hyperbolic groups are finite, F(H) is finite.
• The construction shows that Q ∖ {g} generates a proper subgroup of H, proving g ∉ F(H).
• The proof establishes a uniform lower bound T on the word length of nontrivial elements in the subgroup generated by {h₁, ..., hₜ}, ensuring it is not all of H.
• The result implies that the rank and generation problems in hyperbolic groups cannot be simplified via quotienting by the Frattini subgroup, unlike in nilpotent groups.