Embedding into a finitely presented group

By James Belk, James Hyde, and Francesco Matucci

Abstract

We observe that the group of all lifts of elements of Thompson’s group  to the real line is finitely presented and contains the additive group  of the rational numbers. This gives an explicit realization of the Higman embedding theorem for , answering a Kourovka notebook question of Martin Bridson and Pierre de la Harpe.

Introduction

In 1961, Graham Higman proved that any countable group with a computable presentation can be embedded into a finitely presented group Reference 11. For example, the additive group of rational numbers has computable presentation

and can therefore be embedded into some finitely presented group. Unfortunately, Higman’s construction is difficult to carry out in practice, and group presentations produced by his procedure are quite large and unwieldy.

Higman was for many years interested in finding more explicit embeddings of various naturally occurring recursively presented groups such as into finitely presented groups Reference 15. In 1999, the following question was submitted to the Kourovka notebook Reference 17 and labeled as a “well-known problem”. The question is attributed to Pierre de la Harpe in Reference 17, but Martin Bridson and de la Harpe have informed us that the question was originally submitted jointly by the two of them.

Problem 14.10(a) (Reference 17).

It is known that any recursively presented group embeds in a finitely presented group. Find an explicit and “natural” finitely presented group and an embedding of the additive group of the rationals in .

The problem then asks the same question for the group . The problem originally included a part (b) that asked for any finitely generated example, although such examples had already been supplied by Hall in 1959 Reference 10. In particular, Hall observed that if is a vector space over with basis and are the linear transformations of defined by and , where is some enumeration of the primes, then the orbit of under generates  as an abelian group, and hence the semidirect product is finitely generated and contains . Further finitely generated examples were later supplied by Mikaelian Reference 18. As for embeddings into finitely presented groups, Mikaelian Reference 19 has described how to explicitly carry out Higman’s construction for as well as many other groups of interest, such as .

In this note we observe that embeds into a finitely presented group which was introduced by Ghys and Sergiescu in 1987 Reference 9. This is an explicit group of homeomorphisms of the real line, which has a presentation with two generators and four relators (see Remark 3). Specifically, consists of all homeomorphisms that satisfy the following conditions:

(1)

The homeomorphism is piecewise-linear, with finitely many breakpoints on each compact interval.

(2)

Each linear portion of has the form , where and is a dyadic rational.

(3)

Each breakpoint of has dyadic rational coordinates.

(4)

The homeomorphism commutes with the translation . That is, for all .

It follows from (4) that the set of breakpoints of is invariant under , and hence any is either linear or has infinitely many breakpoints. As a group of orientation-preserving homeomorphisms of the real line, is torsion-free, and indeed right-orderable Reference 8, Theorem 6.8. The monomorphisms that we describe below are order-preserving.

The group is closely related to the three groups , , and introduced by Richard J. Thompson in the 1960s Reference 5Reference 20. Thompson’s group  arises naturally as the “group of associative laws” and also arose independently in homotopy theory Reference 6, while Thompson’s groups and were the first known examples of infinite, finitely presented simple groups. Thompson’s group  is the group of homeomorphisms of the circle that satisfy conditions (1), (2), and (3) above, and  is precisely the group of all “lifts” of elements of to the real line. In particular, the quotient of by the cyclic subgroup generated by is isomorphic to . This cyclic subgroup is precisely the center of , and therefore is a central extension of  (though it is not the universal central extension). Ghys and Sergiescu introduced in this context as part of their investigation into the cohomology of  Reference 9.

Thompson made the surprising observation that contains elements of arbitrary finite order Reference 20. In 2011, Bleak, Kassabov, and the third author gave an elementary argument that embeds into Thompson’s group  Reference 2. They did not consider the consequences for the group , but it follows easily from their result that embeds into . We give a self-contained proof of this below, and indeed we prove something a bit stronger:

Theorem 1.

The group has continuum many different subgroups isomorphic to , all of which contain the center of .

Brin has proven that  embeds naturally into the automorphism group of Thompson’s group . Specifically, Brin proved Reference 4, Theorem 1 that has an index-two subgroup which is isomorphic to the group of all homeomorphisms of  that satisfy conditions (1), (2), and (3) above and agree with elements of in neighborhoods of and . Brin also showed Reference 4, Theorem 1 that this group fits into a short exact sequence

and it follows easily that is finitely presented. Indeed, Burillo and Cleary have computed an explicit finite presentation for in Reference 3. This gives another natural example of a finitely presented group that contains :

Corollary 2.

The automorphism group of Thompson’s group  has a subgroup isomorphic to .

Remark 3.

The smallest known presentation for Thompson’s group has two generators and five relators, and was derived by Lochak and Schneps in Reference 16. (Note that the version in Reference 16 contains a typo. See Reference 7, Proposition 1.3 for a corrected version.) Using this presentation together with the fact that is a central extension of by , it is not difficult to derive a presentation for with two generators and four relators. Specifically,

where and are the elements of whose restrictions to are defined by

Remark 4.

In addition to the above results, we have obtained an explicit embedding of and hence an embedding of  into a finitely presented simple group , verifying the Boone–Higman conjecture in the case of . We have also obtained an explicit finitely presented simple group that contains all countable abelian groups. Both of these groups will be described in a forthcoming paper.

Remark 5.

Hurley Reference 14 and Ould Houcine Reference 13 have proven that there exists a finitely presented group whose center is isomorphic to . It would be interesting to find a natural example of such a group, or at least a natural example of a finitely presented group whose center contains .

Inclusion of into

Let denote the (uncountable) group of all piecewise-linear homeomorphisms of that satisfy conditions (1) through (3) for elements of given in the Introduction. The group is precisely the centralizer in of the homeomorphism .

If and are closed intervals in , we say that a piecewise-linear homeomorphism is Thompson-like if it is a restriction of an element of ; i.e., if it satisfies conditions (1) through (3) for elements of given in the Introduction. It is well-known that if and have dyadic rational endpoints, then there exists at least one Thompson-like homeomorphism (cf. Reference 5, Lemma 4.2).

Lemma 6.

Let be an element of without fixed points, and let . Then there exist infinitely many different such that .

Proof.

Without loss of generality, suppose that . Choose dyadic rationals , and for each choose a Thompson-like homeomorphism . Let be the homeomorphism , and let be the homeomorphism that agrees with on each () and satisfies

for each with .

To prove that , observe that on the interval , the function restricts to the composition

Since , the expression above simplifies to . Thus agrees with on , and it follows easily that . Moreover, since there are infinitely many possible choices for , …, and , …, , there are infinitely many possibilities for .

Lemma 7.

Let , and let so that . Then for every there exist infinitely many different so that .

Proof.

Note that cannot have any fixed points, since these would also be fixed points of . Therefore, by Lemma 6, there exist infinitely many such that . Any such homeomorphism commutes with since , and therefore every such lies in .

Proposition 8.

The group has continuum many subgroups isomorphic to .

Proof.

Observe that has presentation

To obtain an embedding of into it suffices to find a sequence of elements of  such that has infinite order and for all . Such a sequence can be defined recursively by letting and then repeatedly applying Lemma 7 to find, for each , an element such that . Since there are infinitely many choices for  at each stage, this procedure constructs continuum many different copies of .

Remark 9.

Since each subgroup of is conjugate to only countably many other subgroups, it follows from Proposition 8 that has continuum many conjugacy classes of subgroups isomorphic to .

Remark 10.

The choice of the elements in the proof of Proposition 8 can be carried out constructively. For example, let be the decreasing sequence of dyadics in defined recursively by and . Let , and for each let be the th root of in that satisfies

Then the sequence generates a subgroup of isomorphic to .

It is possible to write these elements explicitly in terms of the generators for given in Remark 3. Specifically, let be any element of which is the identity on and has slope on (e.g., ), let be any element of which maps linearly to (e.g., ), and let be the element of which satisfies

Define a sequence of elements recursively by and

for , where denotes . Then maps the left half of linearly to , maps the right half of linearly to the left half of , maps linearly to its right half, and is the identity on . The desired sequence can now be defined recursively by

for , where denotes , , , and .

The idea here is that is roughly the same as on and is the identity elsewhere. Since is a fundamental domain for the action of , we can construct by multiplying together conjugates of by powers of , with the correction factor accounting for the overlap between the initial and the last conjugate . The authors have checked all of the above computations in Mathematica Reference 1.

Remark 11.

The copy of constructed in Remark 10 has the property that the orbit of is dense in . For such a copy, the resulting action of on is conjugate by a homeomorphism of to the usual action of on by translation. However, there are also “exotic” copies of in for which the orbit of is not dense in . For example, we can choose a sequence in with and () such that for all . In this case, the subgroup , , , …› is isomorphic to , but the orbit of under the action of this subgroup does not intersect the interval . It follows that the restricted wreath product embeds into , where Thompson’s group  embeds into as the group of elements that are the identity on . Bleak, Kassabov, and the third author used a similar argument to prove that embeds into  Reference 2, Theorem 1.6.

Remark 12.

Higman proved that elements of Thompson’s group of infinite order do not have roots of arbitrarily large orders Reference 12, Corollary 9.3. It follows that does not embed into , and hence does not embed into , either.

Every copy of obtained from the proof of Proposition 8 contains the center of . Proposition 13 asserts that these are all of the subgroups of isomorphic to .

Proposition 13.

Every subgroup of isomorphic to  contains the center of .

Proof.

Let be a subgroup of isomorphic to . Since does not embed into (see Remark 12), the projection homomorphism cannot be injective on , so must intersect the center of  nontrivially. In particular, must contain for some . Since is isomorphic to , there exists an so that . Since and commute, it follows that , and since is torsion-free, we conclude that , and therefore contains .

Acknowledgments

The authors would like to thank Collin Bleak for many helpful conversations and suggestions about this work. We would also like to thank Matthew Brin and Matthew Zaremsky for their comments on an early draft of this manuscript and Martin Bridson and Pierre de la Harpe for comments on the historical perspective. Finally, we would like to thank an anonymous referee for many helpful comments and suggestions.

About the authors

James Belk is a visiting assistant professor at Cornell University. His research interests include geometric group theory, dynamical systems, and fractal geometry.

James Hyde is an H. C. Wang Assistant Professor at Cornell University. His research interests include geometric group theory, especially groups of homeomorphisms.

Francesco Matucci is an associate professor at the University of Milano–Bicocca. He is interested in questions of asymptotic, combinatorial, and geometric group theory.

Mathematical Fragments

Remark 3.

The smallest known presentation for Thompson’s group has two generators and five relators, and was derived by Lochak and Schneps in Reference 16. (Note that the version in Reference 16 contains a typo. See Reference 7, Proposition 1.3 for a corrected version.) Using this presentation together with the fact that is a central extension of by , it is not difficult to derive a presentation for with two generators and four relators. Specifically,

where and are the elements of whose restrictions to are defined by

Lemma 6.

Let be an element of without fixed points, and let . Then there exist infinitely many different such that .

Lemma 7.

Let , and let so that . Then for every there exist infinitely many different so that .

Proposition 8.

The group has continuum many subgroups isomorphic to .

Remark 10.

The choice of the elements in the proof of Proposition 8 can be carried out constructively. For example, let be the decreasing sequence of dyadics in