We investigate the Tits alternative for cyclically presented groups with length-four positive relators in terms of a system of congruences (A), (B), (C) in the defining parameters, introduced by Bogley and Parker. Except for the case when (B) holds and neither (A) nor (C) hold, we show that the Tits alternative is satisfied; in the remaining case, we show that the Tits alternative is satisfied when the number of generators of the cyclic presentation is at most 20.
The cyclically presented group is the group defined by the cyclic presentation
where is a cyclically reduced word in the free group of rank with generators and is the shift automorphism given by for each (subscripts mod 𝑛). In this article, we study cyclically presented groups , where 𝑤 is a positive word of length four; that is, we study the groups defined by the presentations
( , subscripts mod 𝑛, ). These were first investigated by Bogley and Parker in  in terms of a system of congruences (A), (B), (C) and so-called primary and secondary divisors (defined below). They classify the finite groups and (with two unresolved cases) classify the aspherical presentations . Here we investigate whether the Tits alternative is satisfied; that is, whether each group either contains a non-abelian free subgroup or has a solvable subgroup of finite index. In many cases where we show 𝐺 contains a non-abelian free subgroup, we show that 𝐺 satisfies the stronger properties of being large (that is, it has a finite index subgroup that maps onto the free group of rank 2) or of being SQ-universal (that is, every countable group embeds in a quotient of 𝐺). Similar studies have been carried out for cyclically presented groups with positive relators of length three ([12, 19], with one infinite family of groups unresolved) and with non-positive relators of length three (, with precisely two groups unresolved). Largeness and the Tits alternative have been investigated for other classes of cyclically presented groups in [4, 24].
We prove the following, which shows that the Tits alternative is satisfied, except possibly in the case when both the primary and secondary divisors are equal to one and (B) holds and neither (A) nor (C) hold. We will write as or according to whether the conditions are true or false.
Let , , ,
let , and if , set , and if , set . Suppose that if , then .
If or , then 𝐺 is large.
If , then one of the following holds:
or , in which case 𝐺 contains a non-abelian free subgroup;
, in which case if and , and 𝐺 is large otherwise;
, in which case ;
, in which case 𝐺 is infinite and solvable if and large otherwise;
, in which case 𝐺 is finite and solvable;
and either , in which case , or , in which case .
The existence of an unresolved case in terms of the system of congruences is consistent with the current state of knowledge for the Tits alternative for cyclically presented groups with length-three positive relators, where the Tits alternative is known to be satisfied except for the case where the congruence conditions (A), (B), (C), (D) of  take truth values F, F, F, T, respectively (see [12, 19] for further results concerning that unresolved case).
The case and remains unresolved in general; we show that the Tits alternative also is satisfied in this case when .
Suppose and .
If , then is finite.
If , then is SQ-universal.
We first define the congruences (A), (B), (C) alluded to earlier (throughout this article, congruences are to be taken mod 𝑛, unless otherwise stated):
or or or ,
Then, in the case , we have , and in the case , we have
by cyclically permuting the relators. Therefore, in each case,
As in , we define the primary divisor and the secondary divisor
The shift automorphism 𝜃 of satisfies , and the resulting -action on determines the shift extension , which admits a presentation , where is obtained from 𝑤 by rewriting it in terms of the substitutions (see, for example, [17, Theorem 4]). In particular, the shift extension of is the group
In proving largeness and SQ-universality, we will use the following properties freely (see ). Every large group is SQ-universal. A group that maps homomorphically onto a large group (resp. SQ-universal group) is large (resp. SQ-universal) and if 𝐻 is a finite-index subgroup of a group 𝐺, then 𝐻 is large (resp. SQ-universal) if and only if 𝐺 is large (resp. SQ-universal), so, in particular, is large (resp. SQ-universal) if and only if is large (resp. SQ-universal). A free product (with non-trivial) is large if and only if either 𝐻 and 𝐾 have non-trivial finite homomorphic images such that or either 𝐻 or 𝐾 is large.
We first prove largeness when either the primary or secondary divisor is greater than one.
If the primary divisor , then is large.
The cyclically presented group splits as a free product of 𝑑 copies of the cyclically presented group
(see ). There is an epimorphism 𝜙 of 𝐻 onto given by for each . Therefore, there is an epimorphism of 𝐺 onto the free product of 𝑑 copies of , and hence 𝐺 is large. ∎
If the secondary divisor , then is large.
Introducing the generator and eliminating 𝑥 shows that the shift extension , given at (2.2), has the alternative presentation
The secondary divisor 𝛾 divides each of , so by adjoining the relator , the group maps onto . Therefore, is large if . ∎
Thus we may assume . In Section 3, we build on prior results to show that the Tits alternative is satisfied in the cases where , , , , or . In Section 4, we consider the case and give the proof of Theorem A. In Section 5, we classify the groups that have infinite abelianisation, and observe that if (B) holds and , then the abelianisation is finite. We use this result in Section 6 where we consider the Tits alternative for the case for and prove Theorem B.
3 The cases (F, F, F), (F, T, T), (T, F, F), (T, F, T), (T, T, F), (T, T, T)
Suppose , , and let . Then either , in which case , or , in which case .
Since (A), (B), (C) all hold, all congruences of (A), all congruences of (B) and all congruences of (C) hold. Therefore, . Suppose first that . Then and , so either , in which case implies and then , or 𝑛 is even and , in which case , a contradiction. Suppose then . Then implies 𝑛 is even and . Therefore, , in which case implies , so (by negating subscripts if necessary) which is the group . ∎
Theorem 7.2 of , together with the following technical proposition, deals with the case .
Suppose (B) and (C) hold, and let 𝑝 be as defined at (2.1). Then if and only if and .
By interchanging the roles of , it suffices to consider the case . Then , which divides , so if , we have . For the converse, suppose . Then, by checking each of the congruences in (B) in turn, we see , and hence and . ∎
Corollary 3.3 (to [3, Theorem 7.2])
Suppose . Then the following are equivalent:
𝐺 is finite;
Theorem 8.1 of  deals with the case .
Theorem 3.4 ([3, Theorem 8.1 (b), (c)])
Suppose and . Then is finite and solvable.
We now turn to the cases , , . Recall that the deficiency of a presentation is defined as , and the deficiency of a group 𝐺, , is the maximum of the deficiencies of all finite presentations defining 𝐺.
Suppose or . Then contains a non-abelian free subgroup.
Since (B) and (C) are false, [3, Lemma 6.2] implies that the cyclic presentation satisfies the C(4)-T(4) small cancellation condition and is combinatorially aspherical, and then, by [3, Lemma 6.1 (a)], the group is torsion-free. As discussed in [3, Section 2] (see [2, Section 3], [8, 22]), 𝑃 is therefore topologically aspherical (in the sense that the second homotopy group of the presentation complex of 𝑃 is trivial) if no relator of 𝑃 is a proper power or is conjugate to any other relator or its inverse. Now if a relator is a proper power, then and , and hence (C) holds, a contradiction. Since the relators of are positive words, no relator is conjugate to the inverse of another relator. If a relator is conjugate to a relator ( ), then is freely equal to or or , and by equating subscripts ( ), we see that (C) must hold, a contradiction. Therefore, 𝑃 is topologically aspherical, and hence, by [23, page 478], .
By , a group defined by C(4)-T(4) presentation contains a non-abelian free subgroup unless it is isomorphic to one of 8 groups, each of which either contains non-trivial torsion or has positive deficiency. Therefore, contains a non-abelian free subgroup, as required. ∎
Suppose , and let . If , then , which is infinite and solvable, and 𝐺 is large otherwise.
If , then and or , so
the fundamental group of the Klein bottle, which is infinite and solvable. So assume . Since (C) holds, by [3, Lemma 5.2], the shift extension has a presentation
where 𝑝 is as defined at (2.1). Since (A) holds, either or , and in the latter case, (C) then implies . Therefore, , which maps homomorphically onto the generalised triangle group
Since (B) does not hold, we have , so the group Δ, and hence 𝐸, is large by [1, Theorem B]. ∎
4 The case (F, F, T)
In this section, we prove the following.
Suppose , and let 𝑝 be as defined at (2.1). If and , then ; otherwise, is large.
We prove this via the following three lemmas.
Suppose , and let 𝑝 be as defined at (2.1). If is not large, then one of the following holds:
and , in which case ;
, where and ;
, where and .
Suppose is not large. Since (C) holds, [3, Lemma 5.2] implies that has a presentation of the form
If is even, then 𝐸 maps onto , which is large, a contradiction. Therefore, is odd. If , then (by adjoining the relator ) 𝐸 maps onto , which is large, a contradiction. Therefore, . Also, for any , the group 𝐸 maps onto . If , then the group , which is large if . If , then is large if by [1, Theorem B]. Thus .
If , then , so , in which case by [3, Theorem 7.2], giving case (a). Thus we may assume , so also , ; in particular, . As discussed in Section 2, since (C) holds, ; then, since , we have , where (see [3, Section 3]). Then
the latter case giving case (c). If , then
which, after replacing by 𝐽, gives case (b). ∎
Suppose is even and 𝐽 is odd. Then is large.
Let . Then
Therefore, we have ; that is, (since 𝐽 is odd), and so
Eliminating in turn and writing then gives
By adjoining the relator , the group 𝐺 maps onto which is large, since .∎
Suppose , , . Then is large.
Let , , where is odd, , and suppose that . Then
Then for each , so eliminating and writing , we have
by writing and ( ), where subscripts are now taken mod 𝑚. For each , multiplying the subscripts by and setting gives
Eliminating and in turn and writing , then gives
which (by adjoining relators , and ) maps onto
which is large for all odd by [1, Theorem B]. ∎
Here, we prove the following theorem, which classifies the groups whose abelianisations are infinite.
Suppose . The abelianisation is infinite if and only if 𝑛 is even and is even.
The abelianisation of a cyclically presented group is infinite if and only if for some , where , where is the exponent sum of in 𝑤 (see, for example, [16, page 77]). For the groups , we have , and so is infinite if and only if for some . If 𝑛 is even and is even, then (since ) precisely one of is even, and so satisfies these conditions.
Taking the complex conjugate gives
or equivalently . Similarly, and . These three equations imply that at least one of is equal to −1, for otherwise , a contradiction. Then, by (5.1), we have , or .
Without loss of generality, we may assume , and so, since , 𝑛 is even. Then , so , and hence , where 𝑚 is the order of 𝜁. Now , so 𝑚 is even, so , and hence , is even, as required. ∎
For use in Section 6, we record the following.
If (B) holds and , then is finite.
6 The case (F, T, F)
The case was observed in  to be the most complex case. We have been unable to determine if the Tits alternative is satisfied in this case for all 𝑛, so in this section, we report results of computations that show it is satisfied for .
The case follows from the results of . Specifically, in the case where , and , the group is isomorphic to one of the following groups: (which is finite and solvable of order 220), , , which are non-isomorphic, finite, non-solvable groups of order . These are the groups (I5), (I6 ), (I6 ) discussed in [3, Section 9]. This proves Theorem B (a), and so we may assume . The following lemma (compare [15, Corollary 14]) shows that, to prove is SQ-universal, it suffices to prove that it is hyperbolic.
Let , and . If is hyperbolic, then it is non-elementary hyperbolic, and hence SQ-universal.
A torsion-free group is virtually ℤ if and only if it is isomorphic to ℤ (see, for example, [18, Lemma 3.2]), so any non-trivial, torsion-free, hyperbolic group with finite abelianisation is non-elementary hyperbolic, and hence SQ-universal by [20, 10]. Therefore, it suffices to show that 𝐺 is non-trivial, torsion-free, with finite abelianisation. The group 𝐺 has finite abelianisation by Corollary 5.2, and it is non-trivial since there is an epimorphism onto obtained by sending each to some fixed generator of .
Since and , the group 𝐺 is not of type (I) or (U) of , and so [3, Theorem 9.2] implies that is combinatorially aspherical. As in the proof of Lemma 3.5, since (C) does not hold, no relator of 𝑃 is a proper power or is conjugate to any other relator or its inverse. Thus, 𝑃 is topologically aspherical, and so (as discussed in the proof of Lemma 3.5) 𝐺 is torsion-free, as required. ∎
It is likely to be a challenging problem to determine in general which of the groups are hyperbolic (compare, for example, [7, 6], which consider hyperbolicity of cyclically presented groups with length-three relators). However, the automatic groups software KBMAG  can be used to show that groups are hyperbolic in particular instances.
Using the isomorphisms amongst the family of groups obtained in [3, Section 3], we wrote a computer program in GAP  to obtain a (potentially redundant) list of 4-tuples that define all isomorphism classes of groups with for which . We then attempted to prove that the corresponding groups are hyperbolic using KBMAG. In the handful of cases where the computation was inconclusive, we proved largeness using Magma . In this way, we obtain the following theorem, from which Theorem B (b) follows by an application of Lemma 6.1.
Let , and suppose , , and let . Then 𝐺 is either hyperbolic or is isomorphic to one of the following groups, each of which is large: , , , , , , , or .
The program described above produced a list of 87 4-tuples . Except in the cases listed in the statement and the cases , , , KBMAG proved the corresponding cyclically presented group to be hyperbolic. (In most cases, the computation completed quickly, but a few were computationally challenging, for example, , and for which KBMAG exhibited geodesic difference machines with 3367, 2839, 4183 states, respectively.) The groups , and have shift extensions
respectively (after writing and applying an automorphism of ). Computations in KBMAG show that each of these shift extensions are hyperbolic, and hence the corresponding cyclically presented groups are hyperbolic.
For the remaining 9 groups, Magma’s largeness functionality shows the existence of a finite index subgroup that maps onto the free group of rank 2, and so are large. The groups and the index of the subgroup produced are as follows: (index 2), (index 6), (index 6), (index 5), (index 5), (index 5), (index 4), (index 3), (index 3). ∎
Let , , , and suppose for some . If , , and , then is SQ-universal.
As mentioned in the introduction, the problem of the Tits alternative for cyclically presented groups 𝐺 with length-three positive relators holds a comparable status, in that the Tits alternative is known to be satisfied, except for the case when and the (A), (B), (C), (D) conditions of  are F, F, F, T, respectively. In this case, the group 𝐺 is isomorphic to , so precisely one one-parameter infinite family of groups remains unresolved. The situation is less clear cut in the case of positive length-four relators, where (if and ) there can be more than one group (up to isomorphism) with the same value of 𝑛.
Based on the evidence provided by Theorem B (b), we conclude by posing the following conjecture.
Let , , . Then is SQ-universal.
Funding statement: S. Isherwood was supported by a University of Essex Doctoral Scholarship.
The authors thank Alastair Litterick for his help performing Magma computations.
Communicated by: Alexander Olshanskii
 W. A. Bogley and G. Williams, Coherence, subgroup separability, and metacyclic structures for a class of cyclically presented groups, J. Algebra 480 (2017), 266–297. 10.1016/j.jalgebra.2017.02.002Search in Google Scholar
 D. F. Holt, KBMAG (Knuth–Bendix in Monoids and Automatic Groups), http://www.warwick.ac.uk/staff/D.F.Holt/download/kbmag2/, 2000. Search in Google Scholar
 D. Macpherson, Permutation groups whose subgroups have just finitely many orbits, Ordered Groups and Infinite Permutation Groups, Math. Appl. 354, Kluwer Academic, Dordrecht (1996), 221–229. 10.1007/978-1-4613-3443-9_8Search in Google Scholar
 E. Mohamed and G. Williams, An investigation into the cyclically presented groups with length three positive relators, Exp. Math. (2019), 10.1080/10586458.2019.1655817. 10.1080/10586458.2019.1655817Search in Google Scholar
 A. Y. Olshanskiĭ, -universality of hyperbolic groups, Mat. Sb. 186 (1995), no. 8, 119–132. Search in Google Scholar
 S. J. Pride, The concept of “largeness” in group theory, Word Problems, II (Oxford 1976), Stud. Logic Found. Math. 95, North-Holland, Amsterdam (1980), 299–335. 10.1016/S0049-237X(08)71343-0Search in Google Scholar
 S. J. Pride, Identities among relations of group presentations, Group Theory From a Geometrical Viewpoint (Trieste 1990), World Scientific, River Edge (1991), 687–717. Search in Google Scholar
© 2022 Walter de Gruyter GmbH, Berlin/Boston
This work is licensed under the Creative Commons Attribution 4.0 International License.