$\mathbb{Z}^2$-free subgroups of Thompson's $T$ are virtually free

Abstract: It is an open problem to determine whether surface groups can embed in Thompson's group $V$. We prove that any finitely generated subgroup of Thompson's group $T$ without abelian groups of arbitrary high rank is either virtually abelian or virtually free. In particular, surface groups don't embed in $T$, answering a question of Belk and Moore.

• The paper proves a trichotomy: every finitely generated subgroup of Thompsons T is virtually abelian, virtually free, or contains a copy of the wreath product Z wr Z.
• The authors combine circle dynamics, the GubaSapir theorem, rotation numbers, and quasi-tree methods to show that Z wr Z-free subgroups must be virtually abelian or virtually free.
• The result rules out embeddings of closed higher-genus surface groups in T, while the analogous surface-group question for Thompsons V remains open.

Overview and main result

This paper by Matte Bon and Triestino studies finitely generated subgroups of Thompson's group $T$, the group of orientation-preserving piecewise-linear homeomorphisms of the circle with dyadic slopes and breaks. The main theorem establishes a trichotomy: every finitely generated subgroup $G \le T$ is either virtually abelian, virtually free, or contains a copy of the wreath product $\mathbb{Z} \wr \mathbb{Z}$. Equivalently, any finitely generated $\mathbb{Z} \wr \mathbb{Z}$-free subgroup of $T$ is virtually abelian or virtually free. Since surface groups contain no such wreath products and are neither virtually abelian nor virtually free, this immediately implies that closed higher-genus surface groups do not embed in $T$, answering a question attributed to Belk and Moore. The broader question of whether surface groups embed in $V$ — posed by Bleak, Matucci, and Neunhöffer — remains open.

The result is sharp in a precise sense: Álvarez et al. showed that every finitely generated free-by-finite-cyclic group embeds abstractly into $T$, so the virtually free alternative in the trichotomy is exactly realized. On the other hand, the authors note that an analogous statement for $V$ cannot hold in this form: $V$ contains finitely generated, even finitely presented, $G \le T$0-free subgroups that are not virtually free, such as Houghton's groups $G \le T$1 and $G \le T$2, although these do contain finite-by-cyclic wreath products $G \le T$3. The paper also contrasts $G \le T$4 with the full group $G \le T$5, which does contain surface groups via Fuchsian actions with maximal Euler class constructed by Ghys.

Subgroups with free orbits

The first ingredient is a criterion for virtual freeness valid in Thompson's group $G \le T$6. Using Thurston's interpretation of $G \le T$7 as a group of piecewise-$G \le T$8 bijections of $G \le T$9, the authors prove that if a finitely generated $\mathbb{Z} \wr \mathbb{Z}$0 admits a free orbit on $\mathbb{Z} \wr \mathbb{Z}$1, then $\mathbb{Z} \wr \mathbb{Z}$2 is virtually free.

The proof is geometrically clean. One enlarges the local pieces of generators to generate all of $\mathbb{Z} \wr \mathbb{Z}$3. For any point $\mathbb{Z} \wr \mathbb{Z}$4, the Schreier graph of the orbit $\mathbb{Z} \wr \mathbb{Z}$5 admits a 1-Lipschitz embedding into the Schreier graph of $\mathbb{Z} \wr \mathbb{Z}$6, which is a quasi-tree. A remark credited to Romain Tessera supplies the key shortcut: 1-Lipschitz embeddings preserve boundedness of separation profiles in the sense of Benjamini–Schramm–Timár, and by a theorem of Hume and Mackay, a vertex-transitive bounded-degree connected graph has bounded separation profile if and only if it is a quasi-tree. Taking $\mathbb{Z} \wr \mathbb{Z}$7 with trivial stabilizer identifies the Schreier graph with the Cayley graph of $\mathbb{Z} \wr \mathbb{Z}$8, and a finitely generated group whose Cayley graph is a quasi-tree is virtually free.

The authors observe that the intermediate statement — Schreier graphs of finitely generated subgroups of $\mathbb{Z} \wr \mathbb{Z}$9 acting on the circle are quasi-trees — coincides with Theorem A of Hyde, Skipper, and Zaremsky, obtained independently; the present proof is shorter owing to Tessera's observation. They also compare with Bennett and Bleak, who reach the same conclusion under the stronger hypothesis of a free orbit on an open interval, using the Muller–Schupp characterization of virtually free groups via context-free word problems.

Wreath-free groups of PL circle homeomorphisms

The second stage analyzes $\mathbb{Z} \wr \mathbb{Z}$0-free subgroups of $\mathbb{Z} \wr \mathbb{Z}$1 using circle dynamics. The starting point is a classical theorem of Guba and Sapir: any non-abelian subgroup of $\mathbb{Z} \wr \mathbb{Z}$2 contains a copy of $\mathbb{Z} \wr \mathbb{Z}$3. Consequently, in a $\mathbb{Z} \wr \mathbb{Z}$4-free subgroup of $\mathbb{Z} \wr \mathbb{Z}$5, every point stabilizer is abelian.

The paper then splits according to whether the action is elementary (i.e., preserves a Borel probability measure). For elementary finitely generated $\mathbb{Z} \wr \mathbb{Z}$6-free subgroups, the rotation number homomorphism is used as follows:

• If some element has irrational rotation number, Herman's PL version of Denjoy's theorem gives a unique invariant measure of full support; since elements in the kernel of the rotation number fix its support, the homomorphism is injective and $\mathbb{Z} \wr \mathbb{Z}$7 is abelian.
• If all rotation numbers are rational, finite generation makes the image finite, and the kernel has global fixed points, hence is abelian by the Guba–Sapir corollary. Thus $\mathbb{Z} \wr \mathbb{Z}$8 is virtually abelian.

For non-elementary countable $\mathbb{Z} \wr \mathbb{Z}$9-free subgroups, the authors prove that outside a countable subset of the minimal set $T$0, every $T$1-orbit is free. The argument shows that if a non-trivial element fixed points of $T$2 in its interior, one could find disjoint intervals meeting $T$3 and apply Antonov's theorem to obtain an element $T$4 whose support lies inside the fixed set of $T$5; then $T$6 and $T$7 generate either a copy of $T$8 or violate abelianness of stabilizers. This yields uncountably many free orbits.

Proof of the main theorem

The assembly is short. Let $T$9 be finitely generated and $T$0-free. If the action on $T$1 is elementary, Proposition on elementary subgroups gives that $T$2 is virtually abelian. If non-elementary, the free-orbit proposition provides uncountably many free orbits, and the quasi-tree criterion forces $T$3 to be virtually free. Both alternatives are realized: virtually cyclic groups act elementarily, while the Álvarez et al. embedding theorem realizes the virtually free case.

An implication worth emphasizing: the obstruction to exotic subgroup structure in $T$4 is entirely dynamical. The Guba–Sapir phenomenon (non-abelian interval actions force $T$5) combined with Herman's rigidity of irrational rotations leaves no room for subgroups of intermediate type, in contrast with $T$6 at large, where Fuchsian surface-group actions exist.

Limitations and open questions

The paper is explicit about where its methods stop. First, the extension to Thompson's group $T$7 appears substantially harder: the dynamical arguments rely on the circular order and minimality theory available for actions on $T$8, and the authors note that the analogous approach "seems sensibly more challenging" for $T$9. Whether surface groups embed in $V$0 — the original motivation via Bleak–Matucci–Neunhöffer Question 7 — remains open. Second, the trichotomy is stated only for finitely generated subgroups; the behavior of infinitely generated $V$1-free subgroups of $V$2 is not addressed. Third, the classification of which virtually free groups arise as subgroups of $V$3 (as opposed to $V$4) rests on the external result of Álvarez et al., namely that these are exactly the free-by-finite-cyclic groups.

Conclusion

The paper proves that finitely generated subgroups of Thompson's group $V$5 satisfy a clean trichotomy — virtually abelian, virtually free, or containing $V$6 — thereby ruling out surface group embeddings in $V$7. The proof combines a separation-profile/quasi-tree argument for orbits in $V$8 with classical results of Guba–Sapir, Herman, and Antonov for PL circle dynamics. The corresponding question for $V$9 remains open and constitutes the natural next target for these methods.