Dehn filling Dehn twists
Let be the genus– oriented surface with punctures, with either or . We show that is acylindrically hyperbolic where is the normal subgroup of the mapping class group generated by powers of Dehn twists about curves in for suitable .
Moreover, we show that in low complexity is in fact hyperbolic. In particular, for , we show that the mapping class group is fully residually non-elementary hyperbolic and admits an affine isometric action with unbounded orbits on some space. Moreover, if every hyperbolic group is residually finite, then every convex-cocompact subgroup of is separable.
The aforementioned results follow from general theorems about composite rotating families, in the sense of [Dah18], that come from a collection of subgroups of vertex stabilisers for the action of a group on a hyperbolic graph . We give conditions ensuring that the graph is again hyperbolic and various properties of the action of on persist for the action of on .
- 1 Composite projection systems and rotating families
- 2 Composite projection graphs
- 3 Shortenings and their applications
- 4 Lifting and projecting
- 5 Applications to mapping class groups
Thurston’s Dehn filling theorem has an algebraic counterpart in the context of relatively hyperbolic groups [Osi07, GM08], which has numerous important applications such as in the proof of the Virtual Haken conjecture [Ago13] and in the solution of the isomorphism problem for certain relatively hyperbolic groups [DG18, DT19]. A Dehn filling of a relatively hyperbolic group is the quotient of by the normal closure of normal subgroups of its peripheral subgroups , and the Dehn filling theorem for relatively hyperbolic groups says that such quotients are still relatively hyperbolic provided that the are sufficiently “sparse”. Mapping class groups are not non-trivially relatively hyperbolic except in very low complexity [AAS07, BDM09], but it is still natural to think of their subgroups generated by Dehn twists around curves in a pants decomposition as peripheral subgroups. Hence, we think of the following theorem (Theorem 5.2 below), as a Dehn filling theorem for mapping class groups:
Suppose or , and consider , an oriented surface of genus with punctures. There exists a positive integer so that for all non-zero multiples of , if denotes the normal subgroup of the mapping class group generated by all powers of Dehn twists, then the group is acylindrically hyperbolic.
Residual properties of mapping class groups in low complexity
Recall that the complexity of is defined as . In low complexity, we can press a bit further and study residual properties of mapping class groups. First, we produce many hyperbolic quotients. Recall that, given a class of groups , a group is fully residually if for every finite set there exists a group in and a surjective homomorphism with .
Suppose that . Then is fully residually non-elementary hyperbolic.
It was not previously known whether either of or admits an infinite hyperbolic quotient.
We deduce Theorem 2 from a different statement, Theorem 5.8, which is interesting even in complexity . Specifically, for , this says that is hyperbolic for all suitably large multiples of the constant from Theorem 5.8, and when , the quotient is hyperbolic relative to subgroups isomorphic to with and is therefore again hyperbolic.
From Theorem 2 and results of Yu and Nica (see also Alvarez and Lafforgue), we obtain:
Suppose that . Then admits an affine isometric action with unbounded orbits on some space.
This is related to the question of which mapping class groups have property (T) because, if can be chosen as above so that one could take , we would get an affine isometric action of , with unbounded orbits, on a Hilbert space.
Recall from [FM02] that a subgroup is convex-cocompact if some (hence any) –orbit in the Teichmüller space of is quasiconvex. There are several equivalent characterisations of convex-cocompactness, see [KL08, Ham05, DT15], and one reason this notion is interesting is its connection with hyperbolicity of fundamental groups of surface bundles over surfaces [FM02, Ham05].
In [Rei06], Reid posed the question of whether convex-cocompact subgroups of are separable. Recall that a subgroup is separable if for every there exists a finite group a surjective homomorphism with .
Note that in general contains non-separable subgroups, and in fact this is already the case for [LM07]. (Nonetheless, various geometrically natural subgroups, e.g. curve-stabilisers, are known to be separable in [LM07].)
The techniques of the present paper engage with this question in a somewhat mysterious way:
Assume that all hyperbolic groups are residually finite. Suppose . Then any convex-cocompact subgroup is separable.
The proof of Theorem 4 relies on the hyperbolic quotients of arising in the proof of Theorem 2. The extra work, done in Proposition 5.12, is to show that in all but finitely many such quotients, the subgroup survives as a quasiconvex subgroup, and arbitrary can be separated from in such quotients. At this point, the assumption about residual finiteness is invoked: using a result of [AGM09] (namely, if all hyperbolic groups are residually finite then all hyperbolic groups are QCERF), we then separate the image of from this quasiconvex subgroup in a finite quotient.
More general context and proof strategy
In order to prove our results we actually work in a more general context (that we plan on using in future work, see the next subsection). Roughly speaking, we consider a group acting on a hyperbolic graph so that the vertex set of fits the framework of composite projection systems introduced in [Dah18], and consider quotients of by normal subgroups generated by subgroups of vertex stabilisers consisting of “big rotations”. The main technical innovation we introduce in this paper is our method for proving that the graph is hyperbolic. The strategy is as follows (see Proposition 4.3). Consider a geodesic triangle in . We can lift the 3 sides of the triangle to a concatenation of 3 geodesics, which need not close up. If it doesn’t close up, there is a non-trivial element that maps the initial vertex of the concatenation to the terminal vertex . We then want to change the lifts so that the new concatenation is either a triangle, or at least is “simpler”. What allows us to do this is Corollary 3.6, which we think of as an analogue of the Greendlinger lemma from [DGO17]. What the corollary says is that we have a large rotation around some vertex so that is “simpler”, and needs to lie within distance of one of the lifts. Applying to part of the concatenation yields new lifts with the required property. The measure of complexity of elements of is actually rather complicated, but the same proof strategy works in other contexts as long as there is a version of the Greendlinger lemma so that some notion of complexity gets reduced when applying it. In particular, it can replace the use of the Cartan-Hadamard Theorem (for hyperbolic space) in the context of very rotating families from [DGO17] (thereby making for a simpler proof). On the other hand, Cartan-Hadamard cannot be applied to in our context, and in fact we could not see any way to apply it to any space quasi-isometric to it.
Future work in the hierarchically hyperbolic setting
A natural strategy for extending the above applications beyond complexity involves combining the techniques in the present paper with the theory of hierarchically hyperbolic groups [BHS17b, BHS15] (of which mapping class groups are one of the “type species”).
We believe that the quotients are in fact hierarchically hyperbolic. One could then apply again our results on composite rotating graphs, and take further quotients. The complexity (in the hierarchically hyperbolic sense) decreases at each step, and hierarchically hyperbolic groups of minimal complexity are known to be hyperbolic [BHS17a]. In particular, under the assumption that every hyperbolic group is residually finite, one should be able to prove that for arbitrary , the group admits a non-elementary hyperbolic quotient, and that every convex-cocompact subgroup is separable.
Outline of the paper
In Section 1, we recall the notions of composite projection systems and composite rotating families from [Dah18] and establish some useful facts, relying on the transfer lemma from [Dah18]. The notion of a composite projection system relies on the projection axioms from [BBF15].
In Section 2, we introduce hyperbolic graphs into the picture, and define the notion of a composite projection graph and a composite rotating family on it. Here, we state our main technical result, Theorem 2.1, and give a proof which relies on statements proved in subsequent sections.
The reader mainly interested in the proof of Theorem 2.1 is advised to focus on the main result of Section 3, which is Corollary 3.6; this is the statement, mentioned above, that allows one to lower the “complexity” of elements by applying large rotations. This is used in Section 4 to construct the lifts mentioned above. From the lifting procedure, one obtains the facts used in the proof of Theorem 2.1. In Section 4, we also prove Proposition 4.8, which describes how stabilisers of vertices in intersect . This is not used in the proof of Theorem 2.1, but does play a role in Section 5.
Finally, in Section 5, we consider the case of acting on the curve graph , which is a composite projection graph by Proposition 5.1 (see also [Dah18]). Large powers of Dehn twists generate a collection of rotation subgroups forming a composite rotating family, and we can invoke Theorem 2.1 to obtain Theorem 1. The rest of Section 5 is devoted to the proofs of Theorem 2 and Theorem 4.
1. Composite projection systems and rotating families
We now recall the notion of a composite projection system from [Dah18], and establish some basic facts. The reader familiar with mapping class groups might want to keep in mind that in that context is the collection of (isotopy classes of simple closed) curves, that two curves are active if they intersect, and that is defined using subsurface projection.
1.1. Composite projection systems
Given in a partitioned set , denote by the index such that .
[Dah18, Definition 1.2] Let be the disjoint union of finitely many countable sets . A composite projection system on (or on ), for the constant , consists of
a family of subsets for (the active set for ) such that , and such that if and only if (symmetry in action),
and a family of functions , satisfying:
Symmetry: for ;
Triangle inequality: for all ;
Behrstock inequality: whenever both quantities are defined;
Properness: for all ;
Separation: for ;
Closeness in inaction: if then, for all , we have ;
Finite filling: for all , there is a finite collection such that covers .
By [BBF15, Theorem 3.3], for each , and , and for a suitable choice of , there exists a modified function , satisfying the monotonicity property of [BBF15, Theorem 3.3] (see also [BBFS17, Axiom (SP3)] for a strengthened property).
This function is unfortunately not defined on . However, is defined on all of . Therefore, we define as follows. Let . If , we let , and otherwise, we let .
Let (the set of -large projections between and in the -coordinate). The elements need not be in the same coordinate.
We now introduce the first of various constraints on the constants that will appear. Fix a composite projection system with constant . Let be the constant provided by applying Theorem 3.3 of [BBF15] to each to obtain the maps as above. In particular, now has the monotonicity property: if , then where . Within , the maps continue to be symmetric and satisfy the properness property, with replacing . The same theorem also provides a constant so that, for all pairwise distinct , we have
We emphasise that the constants have been chosen so that the above properties hold within each .
1.2. Composite rotating families
We now recall the notion of a composite rotating family. The main idea to keep in mind is that we want to consist of “large rotations” around , where is thought of as the angle at .
(Composite rotating family) Consider a composite projection system endowed with an action of a group by isomorphisms, i.e. acts on , preserving the partition , and satisfying for all and . Moreover, suppose that if are such that is defined, then for all .
A composite rotating family on , with rotating control is a family of subgroups such that
for all , is an infinite group;
acts by rotations around (i.e. whenever or , the subgroup fixes and ), with
proper isotropy (i.e. for all , the set is finite);
for all , and all , we have ;
if then and commute;
for all and for all , if , then
for all .
Standing assumptions 1.4.
Recall the useful transfer lemma, which allows one to reduce to “transfer” various configurations to a single coordinate.
(Transfer Lemma, [Dah18, Lemma 1.4, Prop. 1.6]) Let . For all , there exists , such that
for all , and all , one has , and
for all that is -active, for all but finitely many elements of , one has ,
there is so that for all that is -active, we have either , or .
Moreover, if has a fixed point in , we can choose to be such a fixed point.
Notice that by the first and third bullets, we have (regardless of which case from the third bullet applies).
Let so that is -active, and let . Then .
Also, the transfer lemma allows one to transfer properness, from Definition 1.1, which will be useful.
For all and all the set is finite.
Assume it is infinite, and extract an infinite family of elements of the same coordinate . We use Lemma 1.5 to transfer and in : for and as in the transfer lemma, we have that for all , and for either the specific element from the lemma (third point), or the identity, (and similarly), one has . One may extract an infinite family of elements for which the are all equal, and the are all equal. This provides two elements and of such that is infinite, which is a contradiction with properness. ∎
Given a composite rotating family, let be the subgroup generated by all the subgroups .
2. Composite projection graphs
We say that is a -composite projection graph if
hyperbolic graph and acts on by simplicial automorphisms.
Composite projection system: has the structure of a composite projection system on which acts by isomorphisms. We let be the constant from Definition 1.1 and let be the constants, depending on , from the discussion following that definition.
Bounded geodesic image (BGIT): There exists so that the following holds. For each so that is defined and larger than , on any geodesic there exists a vertex with .
Moreover, is a composite rotating family with constant on the -composite projection graph if:
is a composite rotating family on a composite projection system , with constant .
fixes any with .
Our main technical statement is about properties of the action of on that persist for the action of on when the rotations are sufficiently large.
Recall that, given a group acting on a metric space , the element is WPD (weakly proper discontinuous) if for every and there exists so that for all the set
Our main goal in this section is to prove Theorem 2.1. The proof refers to various statements which are postponed to subsequent sections, so that the high-level strategy is made clear before the technicalities are introduced.
Let be a -composite projection graph. If is sufficiently large, in terms of , then for , we have:
If the action of on has a loxodromic then so does the action of on . If the action of on has a WPD element, then so does the action of on . If the action of on is non-elementary, then so is the action of on .
We will refer to three facts established in the next section, Corollary 4.5, Proposition 4.6, and Lemma 4.7. Let be the constants from above, which depend on the composite projection system, and recall that is the BGIT constant associated to the composite rotating family. Suppose that satisfies .
Corollary 4.5 implies that is hyperbolic.
Suppose that acts loxodromically on . Suppose, moreover, that there exists such that for all and all for which the preceding quantity is defined. Then Proposition 4.6 ensures that the image of is loxodromic on , and, moreover, if acts on as a WPD isometry, then acts on as a WPD isometry. We need to show that for a suitably large , there exists a loxodromic (resp. loxodromic WPD) element with the desired small-projection property from Proposition 4.6.
Fix a base vertex and let be loxodromic on (we choose it loxodromic WPD if there is such an element in ). We now show that is uniformly bounded whenever it is defined. This is ultimately a consequence of BGIT, which implies that the “tails” of the orbit of do not affect projection distances very much.
Let be the supremum over all , and over all for which the quantity is defined, of . We claim that .
Indeed, let be the hyperbolicity constant for . Then there exists such that each geodesic lies at Hausdorff distance at most from the (quasigeodesic) sequence .
Fix . By BGIT, either , or is adjacent to some vertex of . Hence there exist integers with and depending only on , so that is not adjacent to a vertex of or unless . Thus , by BGIT and the triangle inequality. It follows that . Now, either , in which case , or is one of finitely many elements for which , by Lemma 1.9. Letting be the maximum of over these finitely many elements gives . Hence for all .
Now let , where the infimum is taken over the set of that are loxodromic on . Note that depends only on the composite projection system, its associated constants, the BGIT constant , and the –action, but not on the choice of rotation subgroups. Suppose that . Then, any loxodromic with satisfies the hypothesis of Proposition 4.6 and thus has image which is loxodromic on , and WPD if is itself WPD.
Finally, suppose that is as above and that is a loxodromic element that is independent of and has the property that is also loxodromic on . (If contains independent loxodromics , then we can choose to be a conjugate of by a sufficiently high power of , and see from the above argument that is again loxodromic on .)
Given such a pair , let be the supremum, over all and all where the following quantity is defined, of . We claim that . Indeed, since are independent loxodromics, there exists such that we have the following. For all , any geodesic lies at Hausdorff distance from
We can now argue as above, using BGIT and replacing by .
Now, letting vary over all pairs of independent loxodromic elements of such that are loxodromic on , take . Suppose that . Then any pair of independent loxodromics such that are loxodromic and has the property that are independent loxodromics, by Lemma 4.7. Hence, if , the action of on is non-elementary. ∎
3. Shortenings and their applications
We work in the setting of Section 2, keeping all notation.
The results of this section support the lifting procedure developed in Section 4, which is vital for proving the statements (Proposition 4.6, Corollary 4.5, Lemma 4.7) used in the proof of Theorem 2.1.
The main statements are Corollary 3.6 and Proposition 3.5, on which the corollary depends. In fact, the reader interested in the proof of Theorem 2.1 is advised to read the statement of Corollary 3.6 and then proceed to Section 4. In order to understand the statement of Corollary 3.6, one needs to know the following. For each , there is an associated complexity , where is a countable ordinal and , and is a constant depending on , , and . (This value is one of the sources of the “sufficiently large” constraint on in Theorem 2.1.)
3.1. Structure of the kernel and complexity of elements
For any countable ordinal there exists a subset of such that, denoting by the subgroup of generated by , we have:
if is not a limit ordinal, there exists , and a subset , such that is an amalgamated free product of with the groups
for . ([Dah18, Lem. 2.16])
Notice that each element in the -orbit of naturally corresponds to a vertex in the Bass-Serre tree of the previous decomposition; from now on we implicitly identify any such element with the corresponding vertex.
Suppose that is not a limit ordinal, and let be the Bass-Serre tree of the previous decomposition. Also, let be three vertices of in the -orbit of the vertices of , with in . Then, when seen as elements in , one has , and there exists such that . ([Dah18, Lem. 2.16 with Prop 2.13])
if is a limit ordinal, is the direct union of . ([Dah18, Lem. 2.17])
. ([Dah18, Lem. 2.19])
We can now define the complexity of .
Given , let be the smallest ordinal for which is in a conjugate of in . Observe that is never a limit ordinal, by the third point of Theorem 3.1, and if then .
Given , consider the amalgamated free product decomposition of a conjugate of containing , given by the second point of Theorem 3.1.
Consider the cyclic normal form of the conjugacy class , which is either an element of for some , or a cyclic word , where, for all , we have and for some . Let be the length of this cyclic normal form, namely if is conjugate into some for some , and it is for the above otherwise.
Note that if is seen as an element of , the length of its cyclic normal form is , but we do not set this in the notation since this notation is reserved to the amalgam decomposition of . This way, no has been multiply defined. We adopt the convention that .
3.2. Angles and shortenings
The main point of the following proposition is to relate the normal form of with vertices at which one sees a large projection between -translates. The move that will allow us to shorten normal forms can be pictorially described as follows: Consider the axis of , a vertex on the axis, and an element that stabilises and rotates an edge on the axis containing to the other such edge. Then has shorter normal form than .
Proposition 3.5 (Angles and shortenings).
Let be an element of with . Then all of the following hold:
The element is hyperbolic in the tree , and its axis in this tree contains some vertex in the -orbit of .
For all in the axis of and in the -orbit of , there exists such that the cyclic normal form of is strictly shorter than that of .
If is a vertex in the tree , then there exists a vertex in the -orbit of such that lies in the intersection of the interior of the segment of , and of the axis of . Moreover, for all such , one has .
For all , there is a vertex in the -orbit of on the axis of such that .
Suppose that and let . Then either there exists in the -orbit of that is -inactive, and so that has shorter cyclic normal form than , or there exists that is active for and , with the property that .
The first point is a general fact for elements in amalgamated free products. This is also true of the second point, in the specific setting where there is only one orbit of edges around under the action of its stabilizer, as is the case in our situation. The first part of the third point is also general. The distance estimate of the third point follows from Theorem 3.1 (3).
For the fourth point, we can assume that is not a vertex of , for otherwise we can just use the third bullet. Consider the bi-infinite sequence of points for some vertex in the -orbit of on the axis of . This sequence can be thought of as ranging over points of the axis of in or over points of . Notice that for every , by the third point of Theorem 3.1. One can then see, using the Behrstock inequality and induction, that the set is an open interval , for (notice that is defined for each ). Moreover, cannot be equal to , since in that case, for all negative , would be in , contradicting the properness of the projection system. Thus, , while . The Behrstock inequality again ensures that . After translation by , we have .
There are two cases to consider. First, suppose that . In that case, the triangle inequality gives (we used ). The second case is when . Then (we used ). In both cases, we obtained the desired conclusion.
Let us prove the fifth point. The first case is when has no fixed point in . In that case, the conjugate has no fixed point either, since a fixed point for one would give a fixed point for the other by translation by . Then for all (so in particular in the axis of in the tree ), and are -active. We now consider as in the transfer lemma, and we apply the previous point for . There is on the axis of such that . By Remark 1.7, we have .
Notice that , and the latter quantity is again bounded by by Remark 1.7. Hence, we get
Let us now treat the case where has a fixed point in . Pick one fixed point and consider first the case where is not active for one of the vertices of the axis. Then, by the second bullet, one could shorten the length of using the element associated to this vertex in the normal form of , so we are done. We may thus assume that is active for all in the axis. It follows that is active for all in the axis as well, and the argument of the previous case can be applied (in view of the “moreover” part of the transfer lemma). ∎
3.3. Rotations to reduce complexity
For convenience, let .
Corollary 3.6 (Rotating to reduce the complexity ).
For all , and all , there is (here and ) so that in lexicographic order and either
is -inactive, or
and are -active and .
After conjugation by a suitable element and replacing with , one can assume that . If then we consider as in the fourth bullet from Proposition 3.5 and set . The cyclic normal form of is shorter than that of by the second bullet from Proposition 3.5. Otherwise, if with , then by the fifth bullet from Proposition 3.5 we either proceed as above, or we find some -inactive so that has shorter cyclic normal form.
In either case, the cyclic length of the conjugacy class is reduced. Either the result is still greater than , and in that case , or it is reduced to (or ) and is actually conjugate into