The Bass–Jiang group for automorphism-induced HNN-extensions
We use the Bass–Jiang group for automorphism-induced HNN-extensions to build a framework for the construction of tractable groups with pathological outer automorphism groups. We apply this framework to a strong form of a question of Bumagin–Wise on the outer automorphism groups of finitely presented, residually finite groups.
Key words and phrases:HNN-extensions, outer automorphism groups, residually finite groups
2010 Mathematics Subject Classification:20E06, 20E26, 20E36, 20F28
Matumoto proved that every group can be realised as the outer automorphism group of some group [matumoto1989any]. Many authors have refined this result by placing restrictions on the groups and [kojima1988isometry] [gobel2000outer] [droste2001all] [braun2003outer] [BumaginWise2005] [frigerio2005countable] [minasyan2009groups] [logan2015outer] [LoganNonRecursive] [LoganHNN]. Bumagin–Wise asked the following question [BumaginWise2005, Problem 1]:
Question 1 (Bumagin–Wise).
Can every countable group be realised as the outer automorphism group of a finitely generated, residually finite group ?
Our main result is Theorem 1.1, which relates to a strong form of Question 1 where the group is taken to be finitely presented rather than finitely generated. Almost nothing is known in general about the outer automorphism groups of finitely presented, residually finite groups. A group has Serre’s property FA if every action of on any tree has a global fixed point; this is related to the property of not splitting non-trivially as a free product with amalgamation or HNN-extension [trees, Theorem 15].
Suppose is a finitely presented group with Serre’s property FA. Then there exists a finitely presented group such that embeds with finite index into . Moreover, if is residually finite then can be chosen to be residually finite.
Examples of finitely presented groups with Serre’s property FA are [serre1974amalgames], triangle groups [trees], finitely presented groups with Kazhdan’s property T [yasuo1982propertyT], random groups in the sense of Gromov [dahmani2011random], R. Thompson’s groups and [farley2011proof], and the Brin–Thompson groups [kato2015higher].
The proof of Theorem 1.1 has two main ingredients. Firstly, Theorem 1.1 applies a certain framework built in this paper and explained below. Secondly, Theorem 1.1 applies a version of Rips’ construction [belegradek2008rips]. Rips’ construction is usually applied to obtain finitely generated, non-finitely presentable groups with pathological properties; our application to finitely presented groups is unusual.
The framework. This paper principally consists of an in-depth analysis of the Bass–Jiang group for automorphism-induced HNN-extensions. The Bass–Jiang group of an HNN-extension , defined in Section 3, is a subgroup of which arises naturally in the context of Bass–Serre theory. Automorphism-induced HNN-extensions, defined in Section 2, are a class of tractable HNN-extensions; by “tractable” we mean that they are an easy class of groups to work with and possess nicer properties than general HNN-extensions. In Section LABEL:sec:applications we package the technical results of this analysis into a framework for the construction of tractable groups with pathological outer automorphism groups. This framework is applied to prove Theorem 1.1 (see also [LoganNonRecursive] [LoganHNN]).
Describing . Theorem 1.2 is a striking, non-technical special case of the framework described in Section LABEL:sec:applications. This theorem gives a convenient description of for certain residually finite HNN-extensions . As our motivation comes from Question 1, the assumption that is residually finite is completely natural. For functions and write , and write for all . For define . Define as:
and this is a subgroup of .
Let , and . Assume that , that is finitely generated, and that is residually finite. Then we have a short exact sequence:
where is an index-one or -two subgroup of .
Theorem 1.2 follows immediately from Theorems 4.7 and LABEL:thm:trivialcenter. Indeed, Theorem LABEL:thm:trivialcenter classifies when ; these conditions are applicable here but are omitted for brevity. Note that the proof of Theorem 1.1 requires a more general result than Theorem 1.2, as the groups in Theorem 1.1 are not necessarily residually finite.
A note on labels of results. In a previous paper [LoganNonRecursive] we cited certain results from this current paper. The labels of the results in this current paper have subsequently changed. We record the relevant changes here: Theorem A is now Theorem 4.7, Lemma 2.1 is now Lemma 4.1, Lemma 5.2 is now Lemma LABEL:lem:finitecenter, and Proposition 5.3 is now Proposition 2.1.
Outline of the paper. In Section 2 we define automorphism-induced HNN-extensions and explain their “tractability”. In Section 3 we define the Bass–Jiang group and the Pettet group for HNN-extensions. In Section 4 we prove conditions implying that the Bass–Jiang group is the whole outer automorphism group; our main result is Theorem 4.7. In Section 5 we give a description of the Bass–Jiang group for automorphism-induced HNN-extensions; our main result is Theorem 5.6. In Section LABEL:sec:commutativity we prove results relating the structure of the Bass–Jiang group of an automorphism-induced HNN-extension to commutativity in the base group of ; our main result is Theorem LABEL:thm:trivialcenter. In Section LABEL:sec:applications we package the technical results of this paper into a framework for the construction of tractable groups with pathological outer automorphism groups, and we prove Theorem 1.1.
Acknowledgements. The author would like to thank Stephen J. Pride and Tara Brendle for many helpful discussions about this paper.
2. Automorphism-induced HNN-extensions
Let be a group and let be an isomorphism of subgroups of . An HNN-extension of over is a group with relative presentation . We write . An automorphism-induced HNN-extension is an HNN-extension with relative presentation
where and ;111If then is the mapping torus of . Here the Bass–Jiang group (see Section 3) is very different because extends to an inner automorphism [logan2015outer]. the isomorphism of associated subgroups is induced by the automorphism . We write . Throughout this paper we use the letters , , and as above.
In a general HNN-extension the form of the embeddings of the two associated subgroups and may be completely different. For example, may be normal while is malnormal. In contrast, in an automorphism-induced HNN-extension the form of the embeddings of and are necessarily the same (both normal, both malnormal, and so on), and in practice the image group may be disregarded. For example, supposing is finitely generated and residually finite, then is residually finite if and only if is residually separable in [Logan2017Residual]. In this current paper, Theorems 1.2, 5.6 and LABEL:thm:trivialcenter do not mention the image group .
The Baumslag–Solitar groups are the HNN-extensions of the infinite cyclic group with non-trivial associated subgroups. These provide standard examples of HNN-extensions with pathological properties. For example, Baumslag–Solitar groups can be non-Hopfian (they are Hopfian if any only if , or , or [collins1983automorphisms]); they can be Hopfian but non-residually finite (they are residually finite if and only if , or , or [meskin1972nonresidually], and note that a finitely generated, residually finite group is Hopfian); and they can have non-finitely generated outer automorphism group (for example, is not finitely generated [collins1983automorphisms]). However, automorphism-induced Baumslag–Solitar groups have , so are residually finite, by the above classification, and have virtually cyclic outer automorphism group [GHMR].
The above two paragraphs demonstrate that automorphism-induced HNN-extensions are more tractable and in certain cases have nicer properties than general HNN-extensions. Therefore, increasing the complexity of the base group maintains the tractability of these HNN-extensions but can allow for pathological properties; see, for example, [LoganNonRecursive], where is a cubulated hyperbolic group with Serre’s property FA, and [LoganHNN], where is a triangle group.
Notation. We write and we write for the inner automorphism defined by for all . These contrast with the notation used here for HNN-extensions because it makes certain proofs more readable.
The normaliser-quotient . The following proposition underlies this whole paper. It implies that embeds into for , and this paper represents an effort to obtain conditions implying that is “close to” (for example, the are isomorphic, commensurable, and so on).
Proposition 2.1 (Ateş–Logan–Pride [AtesPride]).
Let . For define , . Then the map
is a homomorphism with kernel .
It is routine to prove that . Then is a homomorphism as , and it has kernel as if and only if by Britton’s Lemma [L-S, Section IV.2]. ∎
In the framework built by this paper and explained in Section LABEL:sec:applications, the pathological properties of , where , are inherited from the normaliser-quotient . In particular, by Proposition 2.1, if is not residually finite and is finitely generated then is not residually finite [Baumslag] (and this is an “if and only if” statement if has finite index in [Logan2017Residual]). We see later, in Theorem LABEL:thm:trivialcenter, that if then also embeds into , while Lemma LABEL:lem:finitecenter gives a finite-index version of this embedding when . Thus the properties of are in a certain sense bestowed upon .
Note that if is infinite cyclic (and hence is a Baumslag–Solitar group) then is cyclic, and so bestows no pathological properties upon .
3. The Bass–Jiang group
This paper analyses the Bass–Jiang group for automorphism induced HNN-extensions. The proof of Theorem 1.1 applies this analysis. In this section we define the Bass–Jiang group of an HNN-extension . We also define the Pettet group , which is applied in one of our main technical results, Theorem 4.7. The Pettet group lies between the Bass–Jiang group and the full outer automorphism group:
Note that these groups may be defined for graphs of groups in general [Bass1996automorphism] [pettet1999automorphism], but for brevity we only define them for HNN-extensions.
Bass–Serre theory. The definitions in this section and certain proofs in Section 4 apply Bass–Serre theory. All relevant definitions and notation are taken from Serre’s book [trees]. If then acts in a standard way on a tree , called the Bass–Serre tree of . Vertex-stabilisers of are precisely the conjugates of the base group , so for some , while edge-stabilisers are precisely the conjugates of the associated group , so for some . For vertices we write for the geodesic connecting and , and we define the length of , denoted , to be the number of (positive) edges in .
The Bass–Jiang group. If then denotes the coset of containing . The Bass–Jiang group of is the subgroup of consisting of those which have a representative such that and for some , . It is helpful to have a geometric view of the Bass–Jiang group: If is the full pre-image of in then acts on the Bass–Serre of such that the diagram in Figure 2 commutes.
The Bass–Jiang group for the fundamental group of a graph of groups was described by Bass–Jiang [Bass1996automorphism]. Levitt described a group related to in his investigation of for a one-ended hyperbolic group [levitt2005automorphisms]. Gilbert–Howie–Metaftsis–Raptis gave conditions implying for a generalised Baumslag–Solitar group [GHMR]. Theorem 4.7 mirrors this result of Gilbert–Howie–Metaftsis–Raptis.
The Pettet group. The Pettet group of is the subgroup of consisting of those which have a representative such that . It is helpful to have a geometric view of the Pettet group: If is the full pre-image of in then acts on the Bass–Serre tree of such that the diagram in Figure 2 commutes.
M. Pettet studied , and implicitly the Pettet group [pettet1999automorphism]. He focused on conditions for the Pettet group to equal the Bass–Jiang group; these are related to Conditions (1) and (2) from Theorem 4.7.
Two examples. We now give two examples which demonstrate that, in general, and . Our first example is due to Collins–Levin [collins1983automorphisms] and Levitt [levitt2007GBSautomorphism]. Consider the HNN-extension , with base group . The map is an automorphism of , and clearly . However, and so .
We shall write for the free group with free basis . Consider the automorphism-induced HNN-extension , with base group . Clearly , and hence the map is an automorphism of . However, as is not conjugate to any power of in , so .
4. The Bass–Jiang group versus
The Bass–Jiang group is most of interest when it is the full outer automorphism group , so . In this section we prove Theorem 4.7, which gives conditions implying that . To prove these conditions we use the Pettet group , as defined in Section 3. Throughout the remainder of this paper we assume all HNN-extensions are automorphism-induced, so , unless we explicitly state otherwise. We emphasize that this assumption is necessary for our results.
4.1. Conditions implying
Lemma 4.1 now proves that for , if has Serre’s property FA then the Pettet group is equal to the full outer automorphism group. We then have .
We note the similarity between Lemma 4.1 and a comment of Pettet [pettet1999automorphism, Introduction]. The principal difference is that Pettet is additionally assuming conditions implying ; we require no such preliminary assumptions. The proof of Lemma 4.1 shows that under the conditions of the lemma the base group is always conjugacy maximal in . The proof of Lemma 4.1 may be written purely algebraically; we have used Bass–Serre theory so as to be similar to the other proofs on Section 4. Recall that in an automorphism-induced HNN-extension.
Let . If has Serre’s property FA then .
We prove that if then , where ; this is sufficient as then is a representative for such that . So, let and consider the action of on the Bass–Serre tree of . As has Serre’s property FA, stabilises some vertex of . Hence, with . Similarly, with , and so with .
Suppose that and we shall find a contradiction. By the above, . Now, there exist vertices such that and . If then , a contradiction. So and stabilises the geodesic . In particular, stabilises the initial edge of , and therefore there exists such that or . Hence , a contradiction. ∎
4.2. Conditions implying
We now work towards proving Lemma 4.6, which gives sufficient conditions for the Bass–Jiang group to be equal to the Pettet group of . We then have .
Conjugacy. We begin our proof of Lemma 4.6 with Lemma 4.2, which describes conjugacy in automorphism-induced HNN-extensions. A word in an HNN-extension is a product of the form where , . For words and we write if and are exactly the same word, and we say is -reduced if it contains no subwords of the form , , or , . Every element of an HNN-extension has a representative word which is -reduced, by Britton’s Lemma.
Let . For all there exists and such that if with then .
Note that and are dependent on , not on and .
Let be arbitrary. Let be the vertices of the Bass–Serre tree of such that and . Let be a -reduced word in representing . We induct on , the length of the geodesic . If then and the result trivially holds. For the induction step, note that if then decomposes as and where and . We also have a decomposition , and where is such that and the result holds for . Then:
as required, where and . ∎
Let . Suppose that . Additionally, suppose that there does not exist any such that and there does not exist any such that . Then there exists such that .
Suppose , and let be such that . Recall that acts on the vertices of the Bass–Serre tree of . Let be connected by a single edge , and suppose that and that . Then . Write , so , and consider the geodesic . As , . Note that stabilises because stabilises (or, algebraically, because ), so . Note also that the first edge in this geodesic has stabiliser an -conjugate either of or of , and without loss of generality we can assume is either or .
There exists a edge in the geodesic such that [pettet1999automorphism, Lemma 2.2], so for some . Therefore, . As or , we have that or , so for some , by Lemma 4.2.
Recall that contains more than one edge. Let be adjacent to . Then for some , so
Noting that , we therefore have that every element of is equal to an element of the form , , and hence of the form , with fixed. Hence, , and so . By the assumptions of the lemma , as required, where if or if . ∎
Residual finiteness. We now prove Lemma 4.4, which provides conditions for to hold in a general HNN-extension . Lemma 4.5 is essentially Lemma 4.4 for an automorphism-induced HNN-extension. If the conditions of either of these lemmas hold then the HNN-extension is residually finite.
Baumslag–Tretkoff gave conditions for general HNN-extensions to be residually finite [BaumslagTretkoff], and the conditions of Lemma 4.4 are a strong form of these residual finiteness conditions. A subgroup is characteristic if for all .
Let where is an isomorphism of subgroups of , and let be finitely generated and residually finite. Suppose that for any positive integer and elements () with , , , there exists a characteristic subgroup of finite index in such that
, the empty set, and ,
maps into and maps into .
Note that induces an isomorphism of onto . We write . Then the natural map induces a homomorphism of onto [BaumslagTretkoff, Proof of Theorem 4.2]. Note that is finite, so [pettet1999automorphism, Theorem 1].
Suppose . Then there exists with such that is -reduced with . Note that for each , either or . Therefore, pick and the elements , , such that that each is an or as appropriate, pick characteristic in appropriately and form the corresponding group . Now, induces a map with and .
We now prove that . Firstly, note that as is characteristic in . Then, for all because and are homomorphisms. Hence, is a homomorphism. It is surjective because and generate , and it is injective because is finitely generated and residually finite [BaumslagTretkoff, Theorem 3.1] and hence Hopfian.
Then as and the word is -reduced (as each is an or as appropriate). Hence, , a contradiction. ∎
The only difference between the conditions of Lemma 4.4 and those of Baumslag–Tretkoff is that we require the finite-index subgroup to be characteristic rather than normal. Suppose, as with Baumslag–Tretkoff, we instead assume that is normal in . Then, as is finitely generated, there exists a characteristic subgroup which satisfies the first condition. However, it is not clear that is such that .
Now, if is automorphism-induced then the characteristic subgroup also satisfies the second condition, as for example . This observation gives the following lemma, which corresponds to Condition (3) of Lemma 4.6 (also compare with [BaumslagTretkoff, Lemma 4.4]). A subgroup of is residually separable in if for all there exists a finite index, normal subgroup of , written , such that .
Let , and let be finitely generated and residually finite. If is residually separable in then .
As is residually separable in the conditions of Lemma 4.4 hold. The result follows. ∎
An alternative proof of Lemma 4.5 is as follows: By combining Lemma 4.3 with a result of Shirvani [shirvani1985residually, Lemma 3], we see that the conditions of Lemma 4.5 imply that for some . The result follows from Condition (2) of Lemma 4.6, below (the proof of this condition is independent of the proof of Condition (4)).
Let . Suppose that one of the following holds:
is conjugacy maximal in ;
for some ; or
There does not exist any such that and there does not exist any such that ; or
is finitely generated and residually finite, and is residually separable in ; or
is finitely generated and is residually finite.
If (1) or (2) holds then the result is known to hold [pettet1999automorphism, Lemma 2.6 and Theorem 1]. Now, note that it is impossible for (3) to hold simultaneously to the inequality : if both held then (2) would hold, by Lemma 4.3, and so we obtain , a contradiction. Hence, if (3) holds then the result holds.
4.3. The Bass–Jiang group versus
Theorem 4.7 now gives conditions implying that the Bass–Jiang group is the full outer automorphism group.
Let , and , be an automorphism-induced -extension. If has Serre’s property FA then . If any one of the following statements holds then :
is conjugacy maximal in ;
for some ; or
There does not exist any such that and there does not exist any such that ; or
is residually separable in ; or
is residually finite.
Note that Theorem 4.7 obtains in two disjoint steps; it proves that and separately that . M. Pettet proved a similar result to Theorem 4.7 involving Conditions (1) and (2) [pettet1999automorphism, Theorem 1], but he does not obtain these disjoint steps due to his more general setting.
4.4. A condition implying
We do not know if the conditions of Lemma 4.6 are both necessary and sufficient for to hold, and reading closely the proof of Lemma 4.3 one might suspect that they are. Lemma 4.8 now demonstrates how close the conditions of Lemma 4.6 are to being necessary and sufficient.
Lemma 4.8 demonstrates this by proving that if Conditions (1) and (3) of Lemma 4.6 fail “compatibly” with one another then . Note that these two conditions fail for if and only if, after replacing with if necessary, there exists some such that . The “compatibility” we require is that we can take .
Let . Suppose also that where , . Then .
By replacing with , we may assume . Note that as , and so the word is -reduced. We prove that the map
is an automorphism of ; then and the result follows.
Now, so is a homomorphism. To see that is a surjection we prove that . Begin with the following, where . Note that .
We then have that:
Therefore, is surjective. It is injective because it is invertible, with inverse for all , . ∎
5. Describing the Bass–Jiang group
Bass–Jiang gave a decomposition of the group for a general HNN-extension [Bass1996automorphism]. Their description took the form of a filtration. However, their description contains both of the associated subgroups and , while our description in Theorem 5.6 only contains . The difference is in the description of the kernel from the short exact sequence in Theorem 5.6 ( is formally defined below). Specifically, Bass–Jiang find some such that
In general, . Therefore Bass–Jiang’s decomposition of