Free vs Abelian Tits Alternative

Updated 2 January 2026

The free vs abelian Tits alternative sharply dichotomizes groups: every subgroup either surjects onto a nonabelian free group or is torsion-free abelian with bounded rank.
RAAGs and graph products illustrate the theory, where nonabelian subgroups map onto F2 and abelian subgroups adhere to strict rank limitations.
Geometric tools like Bass–Serre theory, CAT(0) cubulation, and normal form methods underpin proofs and reveal deep insights into subgroup structure.

The free vs. abelian Tits alternative designates a strengthening of the classical Tits alternative, seeking to dichotomize group-theoretic phenomena into the existence of nonabelian free subgroups versus a form of virtual abelianity, often in the strongest sense: every non-abelian subgroup surjects onto a nonabelian free group, and every abelian subgroup is torsion-free and of bounded rank. This contrasts with the original Tits alternative for linear groups, which provides a dichotomy between the existence of nonabelian free subgroups and virtual solvability. The distinction between “free” and “abelian” behaviors and their precise structural and subgroup-theoretic implementations has been clarified in multiple contexts, especially in graph products, right-angled Artin groups (RAAGs), tubular free-by-cyclic groups, Artin groups of FC or two-dimensional type, and pseudofinite groups.

1. Core Definitions and Scope

For a group $G$ , the strongest Tits alternative (abbreviated here as FvA-TA for "Free vs Abelian Tits Alternative") asserts that, up to finite index, every subgroup $S$ of $G$ satisfies:

Either $S$ surjects onto the free group $F_2$ ,
Or $S$ is torsion-free abelian, and often of rank at most $k$ (typically $k=2$ ).

This formulation sharpens both classical statements and most intermediate "strong" or "weak" Tits alternatives. The classical Tits alternative (TA) for finitely generated linear groups over a field states that every subgroup is either virtually solvable or contains a nonabelian free subgroup. The FvA-TA replaces "virtually solvable" with "virtually (finitely generated, torsion-free) abelian" (sometimes with explicit rank bounds), and frequently strengthens "contains" to "surjects onto" regarding free groups (Button, 2015, Martin, 2022, Antolín et al., 2023, Antolín et al., 2011).

2. Structural Realizations in Graph Products and RAAGs

Antolín–Minasyan established that every nonabelian subgroup of a right-angled Artin group admits an epimorphism onto $F_2$ , and every subgroup is either free abelian of finite rank or is large in this sense. For finite graph products of groups, under certain closure axioms on the class of vertex groups, various forms of the Tits alternative are inherited from the vertex groups to the full product. For RAAGs, which are graph products where each vertex group is infinite cyclic, the strongest form holds: every nonabelian subgroup surjects onto $F_2$ , and all other subgroups are free abelian (Antolín et al., 2011).

Table: FvA-TA in Key Group Classes

Group class	FvA-TA holds?	Explicit finite index?	Rank bound on abelian S
Right-angled Artin groups (RAAGs)	Yes	No (every subgroup)	Yes (finite)
Tubular free-by-cyclic groups	Yes	Yes	Yes ( $\leq 2$ )
EAFC Artin groups	Yes	Yes	Yes
Two-dimensional Artin groups	Yes	No	Yes ( $\leq 2$ )
Mapping class groups, infinite type	No	No	No
Pseudofinite groups (various)	Weak forms	N/A	Nilpotent/Abelian cores

For tubular free-by-cyclic groups, a finite index subgroup $H$ exists so that every subgroup $S \leq H$ satisfies $S$ surjects onto $F_2$ or is torsion-free abelian of rank at most $2$ (Button, 2015). This also holds for even Artin groups of FC type after passing to a suitable finite index subgroup (Antolín et al., 2023). Two-dimensional Artin groups admit for all subgroups a dichotomy: each is either virtually free abelian of rank at most $2$ or contains a nonabelian free group (Martin, 2022).

3. Mechanisms: Bass–Serre/Graphed Group Decompositions and Cubulation

Rigorous proofs of the FvA-TA typically employ geometric group theory tools such as Bass–Serre theory, CAT(0) cube complex actions, and analysis of subgroup actions on trees or cube complexes.

For tubular free-by-cyclic groups, construction of “equitable sets” per Wise’s criterion enables free actions on CAT(0) cube complexes, even absent virtual specialness. The subgroup structure then follows from analyzing the quotient of the Bass-Serre tree by subgroup actions: every subgroup either yields a quotient graph of rank at least $2$ (implying a free image) or is abelian with strong restrictions (Button, 2015). For RAAGs and graph products, normal form theory, parabolic closures, and graph-theoretic decomposition underpin proof of the finest dichotomy (Antolín et al., 2011).

In Artin groups of even FC type, induction on the number of generators, explicit analysis of parabolic subgroups (root closure properties), and Bass–Serre decompositions via amalgams are crucial. Every “non-large” subgroup, upon intersection with a finite index subgroup, is forced to be free abelian, and all others admit finite-index surjections to $F_2$ (Antolín et al., 2023).

4. Failures and Limitations

The FvA-TA fails in infinite-type mapping class groups, where explicit embeddings of groups such as $\mathbb{Z} \wr \mathbb{Z}$ (restricted wreath product of $\mathbb{Z}$ with itself) are possible. These groups are neither virtually abelian nor contain nonabelian free subgroups; thus such mapping class groups do not satisfy the strong Tits alternative or even the classical Tits alternative in general (Lanier et al., 2019, Allcock, 2020). However, in these settings, certain rigidities survive for normal subgroups, all of which must contain free subgroups if nontrivial (Lanier et al., 2019).

5. Relationship to Classical Tits Alternative, Amenability, and Extensions

The FvA-TA refines the conclusion of the classical Tits alternative, narrowing the possible "small" subgroups from solvable to virtually abelian, and in some cases to strictly abelian or even further to strict rank bounds. The strongest forms (every nonabelian subgroup surjects onto $F_2$ ) arise in highly "nonpositively curved" algebraic environments or where the combinatorics of subgroup generation is tightly controlled by group geometry.

In pseudofinite groups, a nuanced version appears: an $\aleph_0$ -saturated pseudofinite group either contains a free subsemigroup of rank $2$, or is nilpotent-by-(uniformly locally finite); by further assumptions on bounded rank, a further strengthening to nilpotent-by-abelian-by-(uniformly locally finite) is available (Houcine et al., 2012).

Relatedly, amenability properties dovetail: for such pseudofinite groups, the absence of free subsemigroups implies superamenability, while the absence of free nonabelian subgroups implies (uniform) amenability (Houcine et al., 2012). In topological full groups of minimal Cantor actions, the FvA dichotomy manifests as amenability versus the existence of $F_2$ (Szőke, 2018).

6. Examples and Counterexamples

Subgroups in FvA-TA vs. Classical TA contexts:

Setting	Subgroup types permitted by FvA-TA	Subgroup types possible under only TA
RAAGs	finite-rank free abelian or surjects $F_2$	Virtually solvable, possibly non-abelian
Tubular $F_r \rtimes \mathbb{Z}$	$\{1\}$ , $\mathbb{Z}$ , $\mathbb{Z}^2$ , or surjects $F_2$	(as for RAAGs, always one of two cases)
Infinite-type $\operatorname{Mod}(S)$	Nonabelian free, virtually solvable, $\mathbb{Z} \wr \mathbb{Z}$ , Thompson's $F$ , general countable groups	All (no restriction)

The embedding of Grigorchuk group (torsion, non-free, not virtually abelian) and Thompson’s $F$ into infinite-type mapping class groups demonstrates the breakdown of even the classical dichotomy in these settings (Lanier et al., 2019, Allcock, 2020).

7. Horizons: Rank Bounds and Further Directions

All concrete realizations of the FvA-TA in the above research place stringent restrictions on the rank of possible abelian subgroups, typically showing these must be of rank at most two, a feature tightly linked to the structure of the underlying graph or the complexity of the associated Bass–Serre trees or cube complexes (Button, 2015, Martin, 2022). For example, in two-dimensional Artin groups and even FC-type Artin groups, this rank bound is sharp due to the geometry of the Deligne complex or the explicit construction of amalgamated products.

The FvA-TA is thus a critical organizing principle in the modern landscape of geometric and combinatorial group theory, distinguishing the deepest possible contrast between the algebraic presence of free subgroups and the most constrained regime of abelian subgroup structure. Its applicability hinges on the interplay between combinatorial, topological, and geometric group invariants, and its limitations in infinite-type mapping class groups signal the need for further refinement or for additional rigidity assumptions in broader algebraic settings.

Key References: