Geometry of the word problem for 3-manifold groups
We provide an algorithm to solve the word problem in all fundamental groups of 3-manifolds that are either closed, or compact with (finitely many) boundary components consisting of incompressible tori, by showing that these groups are autostackable. In particular, this gives a common framework to solve the word problem in these 3-manifold groups using finite state automata.
We also introduce the notion of a group which is autostackable respecting a subgroup, and show that a fundamental group of a graph of groups whose vertex groups are autostackable respecting any edge group is autostackable. A group that is strongly coset automatic over an autostackable subgroup, using a prefix-closed transversal, is also shown to be autostackable respecting that subgroup. Building on work by Antolin and Ciobanu, we show that a finitely generated group that is hyperbolic relative to a collection of abelian subgroups is also strongly coset automatic relative to each subgroup in the collection. Finally, we show that fundamental groups of compact geometric 3-manifolds, with boundary consisting of (finitely many) incompressible torus components, are autostackable respecting any choice of peripheral subgroup.
One fundamental goal in geometric group theory since its inception has been to find algorithmic and topological characteristics of the Cayley graph satisfied by all closed 3-manifold fundamental groups, to facilitate computations. This was an original motivation for the definition of automatic groups by Epstein, Cannon, Holt, Levy, Paterson, and Thurston , and its recent extension to Cayley automatic groups by Kharlampovich, Khoussainov, and Miasnikov . These constructions, as well as finite convergent rewriting systems, provide a solution to the word problem using finite state automata. However, automaticity fails for 3-manifold groups in two of the eight geometries, and Cayley automaticity and finite convergent rewriting systems are unknown for many 3-manifold groups. Autostackable groups, first introduced by the first two authors and Holt in , are a natural extension of both automatic groups and groups with finite convergent rewriting systems. In common with these two motivating properties, autostackability also gives a solution to the word problem using finite state automata. In this paper we show that the fundamental group of every compact 3–manifold with incompressible toral boundary, and hence every closed 3-manifold group, is autostackable.
Let be a group with a finite inverse-closed generating set . Autostackability is defined using a discrete dynamical system on the Cayley graph of over , as follows. A flow function for with bound , with respect to a spanning tree in , is a function mapping the set of directed edges of to the set of directed paths in , such that
for each the path has the same initial and terminal vertices as and length at most ,
acts as the identity on edges lying in (ignoring direction), and
there is no infinite sequence of edges with each not in and each in the path .
These three conditions are motivated by their consequences for the extension of to directed paths in defined by , where denotes concatenation of paths. Upon iteratively applying to a path , whenever a subpath of lies in , then that subpath remains unchanged in any further iteration , since conditions (F1-2) show that fixes any point that lies in the tree . Condition (F3) ensures that for any path there is a natural number such that is a path in the tree , and hence for all . The bound controls the extent to which each application of can alter a path. Thus when is iterated, paths in “flow”, in bounded steps, toward the tree.
A finitely generated group admitting a bounded flow function over some finite set of generators is called stackable. Let denote the set of words labeling non-backtracking paths in that start at the vertex labeled by the identity of (hence is a prefix-closed set of normal forms for ), and let be the function that returns the label of any directed path in . The group is autostackable if there is a finite generating set with a bounded flow function such that the graph of , written in triples of strings over as
is recognized by a finite state automaton (that is, is a (synchronously) regular language).
To solve the word problem in an autostackable group, given a word in , by using the finite state automaton recognizing to iteratively replace any prefix of the form with and by (when is not ), and performing free reductions, a word is obtained, and if and only if is the empty word. Hence autostackability also implies that the group has a finite presentation.
Both autostackability and its motivating property of automaticity also have equivalent definitions as rewriting systems. In particular, a group is autostackable if and only if admits a bounded regular convergent prefix-rewriting system , and a group is automatic with prefix-closed normal forms if and only if admits an interreduced regular convergent prefix-rewriting system . (See Section 2.2 for definitions of rewriting systems and Section 2.3 for a geometric definition of automaticity.)
The class of autostackable groups contains all groups with a finite convergent rewriting system or an asynchronously automatic structure with prefix-closed (unique) normal forms . Beyond these examples, autostackable groups include some groups that do not have homological type [9, Corollary 4.2] and some groups whose Dehn function is non-elementary primitive recursive; in particular, Hermiller and Martínez-Pérez show in  that the Baumslag-Gersten group is autostackable.
We focus here on the case where is the fundamental group of a connected, compact 3-manifold with incompressible toral boundary. In  it is shown that if no prime factor of admits Nil or Sol geometry, then is automatic. However, the fundamental group of any Nil or Sol manifold does not admit an automatic, or even asynchronously automatic, structure [12, 3]. Replacing the finite state automata by automata with unbounded memory, Bridson and Gilman show in  that the group is asynchronously combable by an indexed language (that is, a set of words recognized by a nested stack automaton), although for some 3-manifolds the language cannot be improved to context-free (and a push-down automaton). Another extension of automaticity, solving the word problem with finite state automata whose alphabets are not based upon a generating set, is given by the more recent concept of Cayley graph automatic groups, introduced by Kharlampovich, Khoussainov and Miasnikov in ; however, it is an open question whether all fundamental groups of closed 3-manifolds with Nil or Sol geometry are Cayley graph automatic. From the rewriting viewpoint, in  Hermiller and Shapiro showed that fundamental groups of closed fibered hyperbolic 3-manifolds admit finite convergent rewriting systems, and that all closed geometric 3-manifold groups in the other 7 geometries do as well. However, the question of whether all closed 3-manifold groups admit a finite convergent rewriting system also remains open.
In this paper we show that every fundamental group of a connected, compact 3-manifold with incompressible toral boundary is autostackable. The results of  above show that the fundamental group of any closed geometric 3-manifold is autostackable; here, we will show that the restriction to geometric manifolds is unnecessary. To do this, we investigate the autostackability of geometric pieces arising in the JSJ decomposition of a 3-manifold, along with closure properties of autostackability under the construction of fundamental groups of graphs of groups, including amalgamated products and HNN extensions.
We begin with background on automata, autostackability, rewriting systems, fundamental groups of graphs of groups, strongly coset automatic groups, relatively hyperbolic groups, and 3-manifolds in Section 2.
Section 3 contains the proof of the autostackability closure property for graphs of groups. We define a group to be autostackable respecting a finitely generated subgroup if has an autostackable structure with flow function and spanning tree on a generating set satisfying:
Subgroup closure: There is a finite inverse-closed generating set for contained in such that contains a spanning tree for the subgraph of , and for all and , .
–translation invariance: There is a subtree of containing the vertex such that the left action of on gives , and for all the trees are disjoint and the intersection of the trees is the vertex . Moreover, the group action outside of preserves the label of the flow function; that is, for all directed edges of not in (with and ) and for all , the flow function satisfies .
(As above, denotes the directed edge of with initial vertex and label .) The conditions on the tree are equivalent to the requirement that the associated normal form set satisfy for some prefix-closed sets of normal forms for and of normal forms for the set of right cosets . The subgroup closure condition together with Lemma 3.2 imply that the subgroup is also autostackable. If the requirement that the graph of the flow function is a regular language is removed, we say that is stackable respecting . We show that autostackability of vertex groups respecting edge groups suffices to preserve autostackability for graphs of groups.
Theorem 3.5 Let be a graph of groups over a finite connected graph with at least one edge. If for each directed edge of the vertex group corresponding to the terminal vertex of is autostackable [respectively, stackable] respecting the associated injective homomorphic image of the edge group , then the fundamental group is autostackable [respectively, stackable].
We note that for the two word problem algorithms that motivated autostackability, some closure properties for the graph of groups construction have been found, but with other added restrictions. For automatic groups, closure of amalgamated free products and HNN extensions over finite subgroups is shown in [12, Thms 12.1.4, 12.1.9] and closure for amalgamated products under other restrictive geometric and language theoretic conditions has been shown in . For groups with finite convergent rewriting systems, closure for HNN-extensions in which one of the associated subgroups equals the base group and the other has finite index in the base group is given in . Closure for stackable groups in the special case of an HNN extension under significantly relaxed assumptions (and using left cosets instead of right) are given by the second author and Martínez-Pérez in . They also prove a closure result for HNN extensions of autostackable groups, with a requirement of further technical assumptions.
In Section 5 we study the relationship between autostackability of a group respecting a subgroup and strong coset automaticity of with respect to defined by Redfern  and Holt and Hurt  (referred to as coset automaticity with the coset fellow-traveler property in the latter paper; see Section 2.3 below for definitions). More precisely, we prove the following.
Theorem 5.1 Let be a finitely generated group and a finitely generated autostackable subgroup of . If the pair is strongly prefix-closed coset automatic, then is autostackable respecting .
Applying this in the case where is hyperbolic relative to a collection of sufficiently nice subgroups, and building upon work of Antolin and Ciobanu , we obtain the following.
Theorem 5.4 Let be a group that is hyperbolic relative to a collection of subgroups and is generated by a finite set . Suppose that for every index , the group is shortlex biautomatic on every finite ordered generating set. Then there is a finite subset such that for every finite generating set of with and any ordering on , and for any , the pair is strongly shortlex coset automatic, and is autostackable respecting , over .
In particular, if is hyperbolic relative to abelian subgroups, then is autostackable respecting any peripheral subgroup.
In Section 6 we prove our results on autostackability of 3-manifold groups. We begin by considering compact geometric 3-manifolds with boundary consisting of a finite number of incompressible tori that arise in a JSJ decomposition of a compact, orientable, prime 3-manifold. Considering the Seifert fibered and hyperbolic cases separately, we obtain the following.
Proposition 6.1 and Corollary 5.5 Let be a finite volume geometric 3-manifold with incompressible toral boundary. Then for each choice of component of , the group is autostackable respecting any conjugate of .
In comparison, finite convergent rewriting systems have been found for all fundamental groups of Seifert fibered knot complements, namely the torus knot groups, by Dekov , and for fundamental groups of alternating knot complements, by Chouraqui . In the case of a finite volume hyperbolic 3-manifold , the fundamental group is hyperbolic relative to the collection of fundamental groups of its torus boundary components by a result of Farb , and so by closure of the class of (prefix-closed) biautomatic groups with respect to relative hyperbolicity (shown by Rebbecchi in ; see also ), the group is biautomatic.
Combining this result on fundamental groups of pieces arising from JSJ decompositions with Theorem 3.5, together with other closure properties for autostackability, yields the result on 3-manifold groups.
Theorem 6.2 Let be a compact -manifold with incompressible toral boundary. Then is autostackable. In particular, if is closed, then is autostackable.
The second author was partially supported by a grant from the National Science Foundation (DMS-1313559).
2. Background: Definitions and notation
In this section we assemble definitions, notation, and theorems that will be used in the rest of the paper, in order to make the paper more self-contained.
Let be a group. Throughout this paper we will assume that every generating set is finite and inverse-closed, and every generating set for a flow function does not contain a letter representing the identity element of the group. By we mean equality in , while denotes equality in the group . For a word , we denote its length by . The identity of the group is written and the empty word in is .
Let be the Cayley graph of with the generators . We denote by the set of oriented edges of the Cayley graph, and denote by the set of directed edge paths in . By we mean the oriented edge with initial vertex labeled by .
By a set of normal forms we mean the image of a section of the natural monoid homomorphism . In particular, every element of has a unique normal form. For , we denote its normal form by .
Let be a subgroup. By a right coset of in we mean a subset of the form for . We denote the set of right cosets by . A subset is a right transversal for in if every right coset of in has a unique representative in .
2.1. Regular languages
Let be a finite set, called an alphabet. The set of all finite strings over (including the empty word ) is written . A language is a subset . Given languages the concatenation of and is the set of all expressions of the form with . Thus is the set of all words of length over ; similarly, we denote the set of all words of length at most over by . The Kleene star of , denoted , is the union of over all integers .
The class of regular languages over is the smallest class of languages that contains all finite languages and is closed under union, intersection, concatenation, complement and Kleene star. (Note that closure under some of these operations is redundant.)
Regular languages are precisely those accepted by finite state automata; that is, by computers with a bounded amount of memory. More precisely, a finite state automaton consists of a finite set of states , an initial state , a set of accept states , a finite set of letters , and a transition function . The map extends to a function ; for a word with each in , the transition function gives . The automaton can also considered as a directed labeled graph whose vertices correspond to the state set , with a directed edge from to labeled by for each and . Using this model is the terminal vertex of the path starting at labeled by . A word is in the language of this automaton if and only if .
The concept of regularity is extended to subsets of a Cartesian product of copies of as follows. Let be a symbol not contained in . Given any tuple (with each ), rewrite to a padded word over the finite alphabet by where and for all and and otherwise. A subset is called a regular language (or, more precisely, synchronously regular) if the set is a regular subset of .
The following theorem, much of the proof of which can be found in [12, Chapter 1], contains closure properties of regular languages that are used later in this paper.
Let be finite alphabets, an element of , regular languages over , a regular language over , a monoid homomorphism, a regular subset of , and the projection map on the -th coordinate. Then the following languages are also regular:
(Homomorphic image) .
(Homomorphic preimage) .
2.2. Autostackability and rewriting systems
As noted in Section 1, the tree defines a set of prefix-closed normal forms, denoted , for , namely the words that label non-backtracking paths in with initial vertex ; as above, we denote the normal form of by . Since , where denotes projection on the first coordinate, Theorem 2.1 implies that the set is a regular language over .
An illustration of the flow function path associated to the edge in the Cayley graph is given in Figure 1.
The flow function yields an algorithm to build a van Kampen diagram for any word over representing the trivial element of . Writing , diagrams for each of the words are recursively constructed, and then glued along the normal forms. (Prefix closure of the normal forms implies that each path labeled by a normal form is a simple path, and hence this gluing preserves planarity of the diagram.) In particular, for and , the edge lies in the tree if and only if either or , which in turn holds if and only if there is a degenerate van Kampen diagram (i.e., containing no 2-cells) for the word . In the case that is not in , the diagram for is built recursively from the 2-cell bounded by and together with van Kampen diagrams for the edges in the path . See [7, 8] for more details.
Since directed edges in are in bijection with and the set of paths in the Cayley graph based at is in bijection with the set of their edge labels, the flow function gives the same information as the stacking function defined by . Thus the set
is the graph of this stacking function. This perspective will be very useful in writing down the constructions of flow functions throughout the paper.
In  the first two authors show that if is a stackable group whose flow function bound is , then is finitely presented with relators given by the labels of loops in of length at most (namely the relations ). Although it is unknown if autostackability is invariant under changes in finite generating sets, we note that it is straightforward to show that if is autostackable with generating set and , then is autostackable with the generators using the same set of normal forms (see [15, Proposition 4.3] for complete details).
Not every stackable group has decidable word problem; Hermiller and Martínez-Pérez  show that there exist groups with a bounded flow function but unsolvable word problem. A group with a bounded flow function whose graph is a recursive (i.e., decidable) language has a word problem solution using the automaton that recognizes . Thus autostackable groups have word problem solutions using finite state automata.
Autostackability also has an interpretation in terms of prefix-rewriting systems. A convergent prefix-rewriting system for a group consists of a finite set and a subset such that is presented as a monoid by and the rewriting operations of the form for all and satisfy:
Termination. There is no infinite sequence of rewritings
Normal Forms. Each is represented by a unique irreducible word (i.e., one that cannot be rewritten) over .
A prefix-rewriting system is called regular if is a regular subset of , and is called interreduced if for each the word is irreducible over and the word is irreducible over . A prefix-rewriting system is called bounded if there is a constant such that for each there are words such that , and .
The following is a recharacterization of autostackability which is useful in interpreting the results and proofs later in this paper.
Theorem 2.2 (Brittenham, Hermiller, Holt ).
Let be a finitely generated group.
The group is autostackable if and only if admits a regular bounded convergent prefix-rewriting system.
The group is stackable if and only if admits a bounded convergent prefix-rewriting system.
A finite convergent rewriting system for consists of a finite set and a finite subset presenting as a monoid such that the regular bounded prefix-rewriting system is convergent. Thus, Theorem 2.2 shows that autostackability is a natural extension of finite convergent rewriting systems, in which the choice of direction of rewritings of bounded length subwords depends upon the prefix appearing before the subword to be rewritten.
2.3. Automatic groups and coset automaticity
In , Redfern introduced the notion of coset automatic group, as well as the geometric condition of strong coset automaticity (using different terminology), studied in more detail by Holt and Hurt in . We consider strong coset automaticity in this paper.
Let be a group generated by a finite inverse-closed set and let be the corresponding Cayley graph. Let denote the path metric distance in . For any word in and integer , let denote the prefix of of length ; that is, if with each , then if and if .
 Let be a group, let be a subgroup of , and let be a constant. The pair satisfies the –coset –fellow traveler property if there exists a finite inverse-closed generating set of and a language containing a representative for each right coset of in , together with a constant , satisfying the property that for any two words with for some , we have for all . The pair is strongly coset automatic if there exists a finite inverse-closed generating set of and a regular language containing a representative for each right coset of in satisfying the –coset –fellow traveler property for some .
The pair is strongly prefix-closed coset automatic if the language is also prefix-closed and contains exactly one representative of each right coset. Given a total ordering on , the pair is strongly shortlex coset automatic if in addition contains only the shortlex least representative (using the shortlex ordering induced by the ordering on ) of each coset.
A group is automatic if the pair is strongly coset automatic. Prefix-closed automaticity and shortlex automaticity are obtained similarly.
We also consider the 2-sided notions of fellow traveling and automaticity in Sections 2.5 and 5. The group is biautomatic if there is a regular language containing a representative of each element of and a constant such that for any and with , we have for all , where . The adjective shortlex is added if is the set of shortlex least representatives of the elements of .
Holt and Hurt show that in the case that the language is the shortlex transversal (i.e., the shortlex least representatives for the cosets), the coset fellow-traveler property yields regularity of the language.
[18, Theorem 2.1] Let be a subgroup of , and let be a finite inverse-closed totally ordered generating set for . If satisfies the –coset –fellow traveler property over using the shortlex transversal for the right cosets of in , then the pair is strongly shortlex coset automatic.
In their work on strong coset automaticity, Holt and Hurt also describe finite state automata that perform multiplication by a generator in strongly shortlex coset automatic groups. As we note in the next Proposition, their construction works without the shortlex ordering as well. We provide some of the details of their construction (with a slight modification), in order to use them later in the proof of Theorem 5.1. As above, denotes the path metric distance in the Cayley graph . For any radius , let
be the set of vertices in the closed ball of radius in the Cayley graph.
 Let be a subgroup of a group , and suppose that is strongly coset automatic over a generating set of with –coset –fellow traveling regular language of representatives of the right cosets of in . Then for each and , there is a finite state automaton accepting the set of padded words corresponding to the set of word pairs
Regularity of the set together with closure of regular languages under product (Theorem 2.1) implies that the language is also regular. Hence the set of padded words corresponding to the pairs of words in is accepted by a finite state automaton , with state set , initial state , accept states , alphabet , and transition function .
Note that the –coset –fellow traveler property implies that for all in , we have for all , and so we can also write
We construct a finite state automaton as follows. The set of states of is , the initial state is , the set of accept states is , and the alphabet is . The transition function is defined by if and otherwise, and (here if either or is , it is treated as the group identity in the expression ). The language of is . ∎
2.4. Graphs of groups
A general reference to the material in this section, with an algebraic approach together with proofs of basic facts about graphs of groups (e.g., invariance under change of spanning tree, injectivity of the natural inclusion of and , existence of the Bass-Serre tree, etc.), can be found in . A more topological viewpoint on this topic is given in .
Let be a connected graph with vertex set , and directed edge set . Each undirected edge is considered to underlie two directed edges with opposite orientations. For an edge , the symbol denotes the directed edge associated with the same undirected edge as but with opposite orientation. The initial vertex of will be called and the terminal vertex .
A graph of groups is a quadruple , where is a graph, is a collection of groups indexed by , is a collection of groups indexed by subject to the condition that for all , , and is a collection of injective homomorphisms .
Let be a graph of groups and let be a spanning tree of . The fundamental group of at , denoted , is the group generated by the union of all of the groups and the set of edges in whose underlying undirected edge is not in , with three types of relations:
for all ,
for all in and , and
for all and .
The fundamental group of a graph of groups can be obtained by iterated HNN extensions (corresponding to the edges in ) and amalgamated free products (corresponding to edges in ). It is also the fundamental group of the corresponding graph of spaces formed from the disjoint union of Eilenberg-MacLane spaces by adding tubes corresponding to and gluing the tubes using identifications corresponding to the maps and .
2.5. Relatively hyperbolic groups
For a group with a finite inverse-closed generating set , let denote the set of directed paths in the associated Cayley graph. Given , write for the group element labeling the initial vertex of , and for the terminal vertex. Given and , the path is a (-quasigeodesic if for every subpath of the inequality holds, where is the path metric distance in .
Let be a group with a finite inverse-closed generating set and let be a collection of proper subgroups of . Let be the Cayley graph of with respect to . For each index let be a set in bijection with , and let .
The relative Cayley graph of relative to , denoted is the Cayley graph of with generating set (with the natural map from ).
A path in penetrates a left coset if contains an edge labeled by a letter in connecting two vertices in .
An –component of a path in is a nonempty subpath of labeled by a word in that is not properly contained in a longer subpath of with label in .
A path is without backtracking if whenever and the path is a concatenation of subpaths with two –components , then the initial vertices , lie in different left cosets of (intuitively, penetrates every left coset at most once).
Geometrically, the relative Cayley graph collapses each left coset of a subgroup in to a diameter subset.
The following definition is a slight modification of the definition originally due to Farb , which is shown by Osin in [25, Appendix] to be equivalent to both Farb’s and Osin’s definitions of relative hyperbolicity for finitely generated groups. Many other equivalent definitions can also be found in the literature (see for example [20, Section 3] for a list of many of these).
Let be a group with a finite generating set and let be a collection of proper subgroups. is hyperbolic relative to if:
is Gromov hyperbolic, and
given any there exists a constant such that for any and any two –quasigeodesics without backtracking that satisfy and , the following hold:
If is an –component of and the path does not penetrate the coset , then .
If is an –component of and is an –component of satisfying , then and .
Property (1) in Definition 2.9 is sometimes called weak relative hyperbolicity and the property (2) is called bounded coset penetration. The collection is called the set of peripheral or parabolic subgroups. A form of bounded coset penetration for –quasigeodesics is given by Osin in [25, Theorem 3.23]; we record that here for use in Section 5.
We note that if a subgroup of is a peripheral subgroup in a relatively hyperbolic structure for (that is, is hyperbolic relative to and for some ), and if is any element of , then the conjugate subgroup is also a peripheral subgroup in a relatively hyperbolic structure for (namely ). In particular, the isometry preserves both hyperbolicity and bounded coset penetration.
When a finitely generated group is hyperbolic relative to , each of the subgroups is also finitely generated [25, Proposition 2.29]. Since relative hyperbolicity of the pair is independent of the finite generating set for [25, Theorem 2.34], for the remainder of the paper we assume that for any relatively hyperbolic group that generates for all .
[1, Construction 4.1] Let be a path in with label , and write with each and each for some index , such that whenever and is the first letter of , then does not lie in for any . The path has no parabolic shortenings if for each the subpath labeled by the subword is a geodesic in the subgraph . If the path has no parabolic shortenings, let be the path in with label where for each the symbol denotes the letter in representing the element of ; the path is the path derived from .
Antolin and Ciobanu studied geodesics and language theoretic properties of relatively hyperbolic groups in . For example, they show that every finitely generated relatively hyperbolic group has a finite generating set, , such that each isometrically embeds in [1, Lemma 5.3] (in fact every finite generating set can be extended to one with this property). We apply the following results from their paper in Section 5.
Let be a finitely generated group hyperbolic relative to and suppose that and . A finite inverse-closed generating set for is called –nice if
every path derived from a geodesic in is a –quasigeodesic in without backtracking,
for all , and
for every total ordering on satisfying the property that is shortlex biautomatic on (with the restriction of the ordering from ) for all , the group is shortlex biautomatic over with respect to that ordering.
[1, Lemma 5.3, Theorem 7.7] Let be a group with finite generating set that is hyperbolic relative to . Then there are constants and and a finite subset of such that every finite generating set of satisfying is a –nice generating set.
In [12, Theorem 4.3.1], Holt shows that every finitely generated abelian group is shortlex automatic over every finite generating set with respect to every ordering of that set; moreover, the structure is also biautomatic. Combining this with Theorem 2.14 implies that any finitely generated group hyperbolic relative to abelian subgroups is shortlex biautomatic (on a –nice generating set), and hence autostackable.
We review some important facts about 3-manifolds that will be used later in the paper. For background, an interested reader can consult [24, 29, 32]. Let be a connected, compact, orientable, three-dimensional manifold with incompressible toral boundary; that is, the boundary of consists of a finite number of incompressible (i.e., –injective) tori.
A 3-manifold is called prime if whenever is a connected sum , then one of or is homeomorphic to .
Decomposing the compact 3-manifold along a disjoint collection of ’s via the connected sum operation, there is a decomposition , where each of the are prime, which is unique up to reordering. This gives a decomposition of as a free product of the fundamental groups of its prime factors.
If is prime then, using Thurston’s Geometrization Conjecture (proved by Perelman; see, e.g., ), either admits a geometric structure based on one of or , in which case is called geometric, or else contains an incompressible torus.
In the nongeometric case, geometrization says that can be split along a collection of non-isotopic, incompressible, two-sided tori in in such a way that every connected component of (where each is an open regular neighborhood of ), known as a piece, has interior admitting a geometric structure with finite volume; this is commonly referred to as a JSJ decomposition. Moreover each piece in is either Seifert fibered or atoroidal, and again by geometrization, the atoroidal pieces have interior that is hyperbolic. In this nongeometric case the fundamental group decomposes as the fundamental group of a graph of groups whose vertex groups are the fundamental groups of the pieces and whose edge groups are all subgroups corresponding to fundamental groups of the tori in the decomposition.
Now suppose that is a piece from the JSJ decomposition in the nongeometric case, or else that in the geometric case. Then is also a compact 3-manifold with incompressible toral boundary. A collection of subgroups arising from this boundary, one for each free homotopy class of boundary component of , or equivalently one for each conjugacy class of subgroup, is a collection of peripheral subgroups of .
If is a Seifert fibered -manifold with boundary, then is a circle bundle over a two-dimensional orbifold with boundary. Consequently, is an extension of the orbifold fundamental group of that -dimensional orbifold by (see [29, Lemma 3.2] for more details). On the other hand, if the interior of is a finite volume hyperbolic -manifold, then is hyperbolic relative to a collection of peripheral () subgroups (see [13, Theorem 5.1]).
3. Autostackability for graphs of groups
In this section we prove Theorem 3.5, the closure of autostackability under the construction of fundamental groups of graphs of groups, in the case that the vertex groups are autostackable respecting their respective edge groups.
We begin by noting that a small extension of the proof that autostackability is invariant under increasing the generating set [15, Proposition 4.3] yields the following Lemma, which will be useful in our closure proof.
Let be autostackable over a generating set respecting a subgroup , and let be another finite inverse-closed generating set for . Then is also autostackable over respecting .
The set of normal forms over is taken to be the same as the normal form set over , and the flow function on edges with labels in is also unchanged. The flow function maps edges labeled by any letter to paths labeled by . ∎
Let be autostackable over a generating set respecting a subgroup with generating set . Suppose that the set of normal forms is , where is a set of normal forms for over and is a set of normal forms over for a right transversal of in . Suppose further that is regular and prefix-closed. Then
and both and are also regular prefix-closed languages.
Note that since is prefix-closed, the empty word is an element of , and so we also have and .
Let be any word in and write where is the maximal prefix of representing an element of . Let be the normal form for the inverse of . Then the word is in , and so by prefix-closure of this set, as well. Since , and normal forms are unique, then . Hence no word in has a nonempty prefix representing an element in . The rest of the result follows from the closure properties of regular languages. ∎
Next we establish some notation used throughout the rest of this section.
Let be a graph of groups on a finite connected graph with vertex set , basepoint , directed edge set , and spanning tree . Let denote the subset of of directed edges whose underlying undirected edges do not lie in . For each let be a finite inverse-closed generating set for . Let and .
The fundamental group of this graph of groups is a quotient of the free product , where is the free group on the set , satisfying for each ; thus, the set is a finite inverse-closed generating set for . With each of the elements of as representatives of the identity of this group, the set also is a finite inverse-closed generating set for .
Define two functions as follows. Let . For each letter , define to be the initial vertex of and to be the terminal vertex of , and for each vertex and letter , define . Finally, for an arbitrary nonempty word , define to be where is the first letter of , and define to be where is the last letter of .
Let be the monoid homomorphism determined by for all and for all . Given any two vertices , let denote the word in that is the unique path without backtracking in the tree from to . Note that if then .
We construct an “inflation” function by defining on any word , with each , as