On dynamics of on characters
We introduce and study an open set of characters of a nonabelian free group, on which the action of the outer automorphism group is properly discontinuous, and which is strictly larger than the set of discrete, faithful convex-cocompact (i.e. Schottky) characters. This implies, in particular, that the outer automorphism group does not act ergodically on the set of characters with dense image. Hence there is a difference between the geometric (discrete vs. dense) decomposition of the characters, and a natural dynamical decomposition.
Let be the free group on generators. Its automorphism group acts naturally, by precomposition, on for any group . The outer automorphism group acts on the quotient , where is understood to act by inner automorphisms. When is a Lie group we consider instead the character variety , the quotient in the sense of geometric invariant theory.
When is a compact Lie group and , Gelander  showed that this action is ergodic, settling a conjecture of Goldman , who had proved it for . When is non compact the situation is different because there is a natural decomposition of , up to measure 0, into (characters of) dense and discrete representations, and in the cases of interest to us the action on the discrete set is not ergodic, indeed even has a nontrivial domain of discontinuity.
See Lubotzky  for a comprehensive survey on the dynamics of representation spaces, from algebraic, geometric and computational points of view.
We will focus on the case of , where the interior of the discrete set is the set of Schottky representations. In this case one can ask if the action is ergodic, or even topologically transitive, in the set of dense representations, but this turns out to be the wrong question. In particular:
There is an open subset of , strictly larger than the set of Schottky characters, which is invariant, and on which acts properly discontinuously.
In other words, the natural “geometric” decomposition of , in terms of discreteness vs. density of the image group, is distinct from the “dynamical” decomposition, in terms of proper discontinuity vs. chaotic action of .
The subset promised in Theorem 1.1 will be the set of primitive-stable representations (see definitions below). It is quite easy to see that this set is open and invariant, and that the action on it is properly discontinuous (Theorems 3.2, 3.3). Thus the main content of this note is the observation, via a lemma of Whitehead on free groups and a little bit of hyperbolic geometry, that it contains non-Schottky (and in particular non-discrete) elements. This will be carried out in Section 4.
One should compare this with results of Goldman  on the rank 2 case for characters, and with work of Bowditch  on the complex rank 2 case. Bowditch, and Tan-Wong-Zhang , studied a condition very similar to primitive stability; we will compare the two in Section 5.
Bill Goldman and Alex Lubotzky got the author interested in this question, and Lubotzky pointed out the notion of “redundant representations”, whose negation leads (eventually) to the primitive-stable condition. Mark Sapir and Vladimir Shpilrain pointed the author to Whitehead’s lemma and Corollary 4.3. The author is also grateful to Dick Canary, Hossein Namazi and Juan Souto for interesting conversations and perspectives. Juan Souto in particular pointed out the connection to Otal’s work. The referee made a number of helpful and incisive suggestions.
2. Background and notation
In the remainder of the paper we fix and let . We also fix a free generating set of .
Note that can be identified with via once the generators are fixed. The quotient , which we denote or just , is obtained by considering characters of representations, which in our case corresponds to trace functions. This quotient naturally has the structure of an algebraic variety, and differs from the purely topological quotient only at reducible points, i.e. representations whose images fix a point on (see Kapovich  or Morgan-Shalen ). Characters of reducible representations form a subset of measure 0 outside all of the sets we shall be considering, and so we shall be able to ignore the distinction between these two quotients.
There is a natural -invariant decomposition of in terms of the geometry of the action of on . Namely, let denote the (characters of) discrete and faithful representations, and let denote those of representations with dense image in .
is nonempty and open, is closed, and has measure 0.
The idea of the measure 0 statement is this: If is not faithful it satisfies some relation; this is a nontrivial algebraic condition, so defines a subvariety of measure 0. There are a countable number of such relations. If is not discrete, consider the identity component of the closure of in . This is a connected Lie subgroup of and hence is either all of (so is dense), or solvable (i.e. elementary, fixing a point in the Riemann sphere), or conjugate to . The latter cases are again detected by algebraic conditions.
Openness of follows from the Kazhdan-Margulis-Zassenhaus lemma , which furnishes a neighborhood of the identity in in which any set of elements generates either an elementary or an indiscrete group. Generating an elementary group is a nontrivial algebraic condition on (for ), and as above for an indiscrete group not to be dense is also a nontrivial algebraic condition. Hence an open dense subset of consists of dense representations. Now given any with dense image, there exist elements such that , and since is open the same is true for sufficiently close to – hence also generates a dense subgroup.
The fact that is closed follows from Jørgensen’s inequality , or alternately from the Kazhdan-Margulis-Zassenhaus lemma. Lemma 2.1 in fact holds for much more general target groups – see [6, 7] for details.
Note, when is compact is empty, and in this case Gelander proved that acts ergodically on and hence on . Our main theorem will show that, in general, the action on is not ergodic.
A Schottky group (or representation) is one which is obtained by a “ping-pong” configuration in the sphere at infinity . That is, suppose that are disjoint closed (topological) disks in and are isometries such that is the closure of the complement of . Then generate a free discrete group of rank , called a Schottky group. The representation sending is discrete and faithful, and moreover, an open neighborhood of it in consists of similar representations. We let denote the open set of all characters of Schottky representations.
Sullivan  proved that
is the interior of .
(This theorem is not known to hold for the higher-dimensional hyperbolic setting; see §5.)
To obtain a geometric restatement of the Schottky condition, consider the limit set for the action of any discrete group of isometries on , namely the minimal closed invariant subset of . The convex hull in of this limit set is invariant, and its quotient by the group is called the convex core of the quotient manifold. The deep usefulness of the convex core in the study of hyperbolic 3-manifolds was first exploited by Thurston. The Schottky condition on is equivalent to the condition that the convex core of is a compact handlebody of genus (see Marden [33, pp. 98-99] for a discussion and references).
3. Primitive-stable representations
Let be a bouquet of oriented circles labeled by our fixed generating set. We let denote the set of bi-infinite (oriented) geodesics in , as in Bestvina-Feighn-Handel . Each such geodesic lifts to an -invariant set of bi-infinite geodesics in , the universal covering tree (and the Cayley graph of with respect to ).
Let be the boundary at infinity of , or the space of ends of the tree . We have a natural action of on . Each element of can be identified with an -invariant subset of (with the diagonal), i.e. the pairs of endpoints of its lifts. acts naturally on (and in general on -invariant subsets of .).
Equivalently we can identify with the set of bi-infinite reduced words in the generators, modulo shift. To every conjugacy class in is associated an element of named , namely the periodic word determined by concatenating infinitely many copies of a cyclically reduced representative of .
An element of is called primitive if it is a member of a free generating set, or equivalently if it is the image of a standard generator by an element of . Let denote the subset of consisting of for conjugacy classes of primitive elements. Note that is -invariant.
Given a representation and a basepoint , there is a unique map mapping the origin of to , -equivariant, and mapping each edge to a geodesic segment. Every element of is represented by an -invariant family of leaves in , which map to a family of broken geodesic paths in .
A representation is primitive-stable if there are constants and a basepoint such that takes all leaves of to -quasi geodesics.
Note that if there is one such basepoint then any basepoint will do, at the expense of increasing . This condition is invariant under conjugacy and so makes sense for . Moreover the property is -invariant since is -invariant. Primitive-stability is a strengthening of the negation of redundancy, whose relevance was explained to me by Alex Lubotzky (see §5).
Let us establish some basic facts.
If is Schottky then it is primitive-stable.
Primitive-stability is an open condition in .
If is primitive-stable then, for every proper free factor of , is Schottky.
To see (1), note that if is discrete and faithful then is the lift to universal covers of an embedding (and homotopy-equivalence) of into the quotient manifold . If is Schottky then the convex core of is compact and hence its homotopy-equivalence to the image of lifts to a quasi-isometry of the convex hull of the group to . It follows that all leaves in map to uniform quasi geodesics, and in particular the primitive ones.
Next we prove (2). Let be primitive-stable, and fix a basepoint and quasi-geodesic constants as in the definition. Let .
Let be a primitive leaf, with vertex sequence , and let . The condition that is quasi-geodesic is equivalent to the following statement: there exist constants and such that, if is the hyperplane perpendicularly bisecting the segment , then for all separates from , and . This is an easy exercise, and we note that determine , and vice versa.
Now consider a representation close to , and let . Up to the action of there are only finitely many sequences of tree edges of length , and hence the relative position (i.e. up to isometry) of and , over all primitive leaves and all , is determined by the image of a finite number of words of . These images each vary continuously with , and hence for sufficiently close to , we have that the separation and distance properties for the still hold, with modified constants. Hence the primitive leaves are still (uniformly) quasi-geodesically mapped by .
Finally we prove (3). Let be a proper free factor, so that with and nontrivial. Suppose is primitive-stable. If is cyclic, then being Schottky is equivalent to ’s generator having loxodromic image, and this is an immediate consequence of having a quasi-geodesic orbit. Hence we may now assume has rank at least 2. By (2), there is a neighborhood of consisting of primitive-stable elements. Suppose were not Schottky. Since is the interior of by Sullivan’s theorem 2.2, and since is dense in by Lemma 2.1, we can perturb arbitrarily slightly to get a dense representation. Leaving unchanged we obtain with dense. Now let be an infinite sequence of reduced words with . For any generator of , a sequence of elementary automorphisms multiplying by generators of (Nielsen moves) takes to , which therefore is primitive. Note that each is cyclically reduced, so primitive-stability of implies that the axes of are uniformly quasi-geodesically mapped by . But this contradicts the fact that while the length of goes to infinity. ∎
be the set of conjugacy classes of primitive-stable representations. We have shown that is an open invariant set containing the Schottky set. In fact,
The action of on is properly discontinuous.
Let denote the translation length of the geodesic representative of for , and let denote the minimal combinatorial length, with respect to the fixed generators of , of any element in the conjugacy class of (equivalently it is the word length of after being cyclically reduced).
Let be a compact set in . For each we have a positive lower bound for over primitive elements of , and a continuity argument as in part (2) of Lemma 3.2 implies that a uniform lower bound
holds over all in . Now on the other hand an upper bound on this ratio holds trivially for any by the triangle inequality applied to any . Continuity again gives us a uniform upper bound
for (here, one should choose a compact preimage of in , which is easy to do).
Now if satisfies , we apply the inequalities to conclude, for in this intersection, that
The proof is then completed by the lemma below. ∎
For any , the set
has finite image in .
In fact much less is needed; it suffices to have the inequality only for with . Let be generators of , and consider the action of on the tree (its Cayley graph). For any , let denote the distance between the axis of and the axis of . A look at the action on the tree indicates, if , that . Hence, since , we get an upper bound on .
An upper bound on the pairwise distances between the axes of the implies that there is a point on which is a bounded distance from all the axes simultaneously (minimize the sum of distances to the axes, which is proper and convex unless all the axes coincide). After conjugating we may assume that this point is the origin. Now the bound implies a finite number of choices for . ∎
4. Whitehead’s lemma and indiscrete primitive-stable representations
In this section we will prove that is strictly bigger than the set of characters of Schottky representations. In particular we will define the notion of a blocking curve on the boundary of a handlebody, and show
Let be discrete, faithful and geometrically finite with one cusp , which is a blocking curve. Then is primitive-stable.
We’ll see below (Lemmas 4.4 and 4.5) that blocking curves are a non-empty class. Using the deformation theory of hyperbolic 3-manifolds one can then obtain primitive-stable points in the boundary of Schottky space – see the end of the section for details.
Note that, since is open, this implies the existence of a rich class of primitive-stable representations, including dense ones (by Lemma 2.1), as well as discrete faithful ones with degenerate ends (since these are topologically generic in the boundary of Schottky space – see  and [12, Cor. A]).
First let us recall Whitehead’s criterion and define the notion of blocking. Whitehead studied the question of which elements in the free group are primitive. He found a necessary combinatorial condition, as part of an algorithm that decides the question of primitivity.
The Whitehead graph
As before, we fix a generating set of . For a word in the generators and their inverses, the Whitehead graph is the graph with vertices labeled , and an edge from to for each string that appears in or in a cyclic permutation of . For more information see Whitehead [46, 47], Stallings  and Otal .
(Whitehead) Let be a cyclically reduced word. If is connected and has no cutpoint, then is not primitive.
Define the “reduced” Whitehead graph to be the same as except that we don’t count cyclic permutations of . In other words we don’t consider the word where is the last letter of and is the first, so may have one fewer edge than .
Let us say that a reduced word is primitive-blocking if it does not appear as a subword of any cyclically reduced primitive word. An immediate corollary of Lemma 4.2 is:
If is a reduced word with connected and without cutpoints, then is primitive-blocking.
Let us also say that is blocking if some power is primitive-blocking. A curve on the surface of the handlebody of genus is blocking if a reduced representative of its conjugacy class in the fundamental group is blocking (with respect to our given generators).
An instructive example of a blocking curve occurs for even rank, when the handlebody is homeomorphic to the product of an interval with a surface with one boundary component:
Let be a surface with one boundary component. The curve in the handlebody is blocking with respect to standard generators of ; in fact its square is primitive-blocking.
Using standard generators for , the boundary is represented by . is a cycle minus one edge (corresponding to ), and contains the missing edge, and so by Whitehead’s lemma is blocking. ∎
One can construct other blocking curves on the boundary of any handlebody by explicit games with train tracks. We omit this approach, and instead study the relationship of the blocking condition to the Masur Domain in the measured lamination space.
Laminations and the Whitehead condition
Let denote Thurston’s space of projectivized measured laminations on the boundary of the handlebody [15, 18]. Within this we have the Masur domain consisting of those laminations that have positive intersection number with every lamination that is a limit of meridians of . This is an open set of full measure in [35, 28].
Using Otal’s work , we can extend Whitehead’s condition to laminations on the boundary as follows. Any free generating set of is dual to a system of disks on , which cut it into a 3-ball (Nielsen). Given such a system of disks and a lamination , realize both and the disk boundaries in minimal position – e.g. fix a hyperbolic metric on and make them geodesics. Otal calls tight with respect to if there are no waves on which are disjoint from . A wave on is an arc properly embedded in , which is homotopic, rel endpoints, through but not through into . In particular if is tight then no arcs of are waves. Hence, if a closed curve is tight with respect to , then its itinerary through the disks describes a cyclically reduced word in with respect to the dual generators.
We define as follows: Cutting along , becomes a planar surface with boundary components, each labeled by or . Let denote the union of the boundary components and the arcs of . Then is obtained from by making the boundary components into vertices and identifying parallel arcs of . In particular is given a planar embedding. If is a single closed curve and is dual to the original generators this is equivalent to the original definition.
Otal proved the following in . We give a proof, since Otal’s thesis is hard to obtain.
(Otal) If , then there is a generating set with dual disks such that is tight with respect to , and is connected and has no cutpoints.
First note that , where runs over meridians of , is positive and realized. For if is a minimizing sequence such that infinitely many of the are distinct then an accumulation point in in will have intersection number 0 with , contradicting The same holds for disk systems, so we may choose a disk system that minimizes .
Now cannot have a wave with respect to . If it did, then a surgery along such a wave would produce a new whose intersection number with is strictly smaller, contradicting the choice of .
If is disconnected then there is a loop in the planar surface which separates the boundary components, and does not intersect (here denotes a regular neighborhood). This gives a meridian that misses in , again contradicting .
If is connected but has a cutpoint, this is represented by a boundary component , equal to one copy of a component of (we are blurring the distinction between the disks in and their boundaries in ). Let denote the other copy. Then separates the planar complex , and we may let be a component of which does not contain . Now cuts into a union of disks-with-holes, one of which, , must contain . Let be the boundary component of that separates from . Any arcs of passing through must pass through (since separates the interior of from the rest of the graph), but arcs of incident to do not meet ; hence is strictly less than .
Because separates from , cutting along and regluing to yields again a connected planar surface – hence is a new disk system, with strictly smaller intersection number with . Again this is a contradiction, so we conclude that is connected and without cutpoints. ∎
Call a lamination blocking, with respect to (or the dual generators), if has no waves with respect to , and there is some such that every length subword of the infinite word determined by a leaf of passing across does not appear in a cyclically reduced primitive word. Note that, for simple closed curves, this coincides with the previous definition of blocking. An immediate corollary of the above lemma is:
A connected Masur-domain lamination (e.g. a simple closed curve or a filling lamination) on the boundary of a handlebody is blocking with respect to some generating set.
Given let be as in Lemma 4.5. In a connected measured lamination every leaf is dense. Thus a sufficiently long leaf of would traverse every edge of , and so the corresponding word is blocking by Corollary 4.3. Note for a simple closed curve this argument shows that its square is primitive-blocking. ∎
Blocking cusps are primitive-stable
We can now provide the proof of Theorem 4.1, namely that a geometrically finite representation with a single blocking cusp is primitive-stable.
Proof of Theorem 4.1.
Let be the quotient manifold, and its convex core. The geometrically finite hypothesis means that is a union of a compact handlebody and a subset of a parabolic cusp (namely a vertical slab in a horoball modulo ) along an annulus with core curve in , which we are further assuming is a blocking curve.
We will prove that all primitive elements of are represented by geodesics in a fixed compact set . The idea is that in order to leave a compact set, a primitive element must wind around the cusp, and this will be prohibited by the blocking property.
Let be a closed geodesic in . Then . The orthogonal projection gives a retraction . Let . Note that if a component of is long then its image winds many times around (in fact the number of times is exponential in the length of ).
Claim: is uniformly quasi-geodesic in , with constants independent of . More precisely, the lift of to the universal cover is uniformly quasi-geodesic with respect to the path metric. This follows from a basic fact about any family of disjoint horoballs in :
Let be a family of disjoint open horoballs in , and let be given by orthogonal projection from to and identity in . If is a geodesic in then is a quasigeodesic in with its path metric, with constants independent of or .
This is closely related to statements in Farb  and Klarreich  and can also be proved in greater generality, e.g. for uniformly separated quasiconvex subsets of a -hyperbolic space. We will sketch a proof for completeness.
Note first, there is a constant such that, if and are horoballs in with , then any two geodesic segments connecting to in their common exterior lie within -neighborhoods of each other.
If is obtained by retracting horoballs to concentric horoballs at depth bounded by , then it suffices to prove the theorem for . This is because any arc on can be approximated in a controlled way by an arc on . Moreover, given it suffices to prove the theorem for the union of horoballs that intersect , since and . We can therefore reduce to the case that any two components of are separated by distance at least 1, and peneterates each component of to depth at least .
Let be a geodesic in with endpoints on . It is therefore a concatentation of hyperbolic geodesics in with endpoints on , alternating with geodesics on in the path (Euclidean) metric. Let be one of the hyperbolic geodesic segments, with endpoints on horoballs . Then is within of the component of connecting these horoballs, and hence can be replaced by an arc traveling along , and of uniformly comparable length. Replacing all hyperbolic segments in this way, and then straightening the arcs of intersection of the resulting path with , we obtain a segment of whose length is comparable with that of Since the endpoints of were arbitrary points of , this gives uniform quasigeodesic constants for . ∎
Now, retracts to a spine of the manifold which can be identified with the bouquet , so is naturally quasi-isometric to the Cayley graph of . In a tree uniform quasi-geodesics are uniformly close to their geodesic representatives.
By Lemma 4.7 is quasi-geodesic in , and it contains a high power of the core curve of . Hence its retraction to the tree contains a high power of some representative of the curve . This subword must be uniformly close to a segment of the form , where is a cyclically reduced representative of . It follows that the geodesic representative of in the tree is uniformly close to a segment of this form as well. Since a tree is 0-hyperbolic, it follows that the geodesic representative of in the tree actually contains a high power of .
But since is blocking, this means that cannot have been primitive. We conclude that all primitive geodesics are trapped in a fixed compact core of . The retraction of this core to the spine of therefore lifts to a quasi-isometry, and stability of quasigeodesics again implies that each primitive geodesic in the tree is uniformly quasi-geodesic. ∎
Proof of Theorem 1.1.
In view of Lemma 3.2 and Theorem 3.3, all that remains to prove is the existence of a non-Schottky primitive stable representation. By Theorem 4.1, any discrete, faithful and geometrically finite representation with a single blocking cusp would do.
Given a curve on the boundary of a handlebody , a sufficent condition that it be realizeable as the single cusp of a geometrically finite representation is that it be homotopically nontrivial in the handlebody, and that any homotopically nontrivial proper annulus in with a boundary component on must be parallel into . This is a consequence of Thurston’s geometrization theorem (see e.g. ) or of Maskit .
Otal  shows that any curve in the Masur Domain has complement satisfying these topological conditions (even more strongly, that is acylindrical, so there are no essential annuli at all in ), and hence can appear as the single cusp of a geometrically finite representation. By Lemma 4.6 such a curve is also blocking with respect to some generating set, and hence gives us a primitive-stable representation.
When is even the example of Lemma 4.4, where , also suffices, even though is not in the Masur domain. Note that we can explicitly construct Fuchsian representations for which is the unique cusp, and Lemma 4.4 provides the blocking property.
5. Further remarks and questions
Having established that acts properly discontinuously on , and that is strictly larger than , one is naturally led to study the dynamical decomposition of . In particular we ask if is the maximal domain of discontinuity, and what happens in its complement. We have only partial results in this direction.
The polar opposite of the primitive-stable characters are the redundant characters , defined (after Lubotzky) as follows: is redundant if there is a proper free factor of such that is dense. Note that is -invariant. Clearly and are disjoint, by Lemma 3.2.
The set of representations with dense image is open (Lemma 2.1), and it follows (applying this to the factors) that is open. The action of on cannot be properly discontinuous, and in fact there is a larger set for which we can show this.
Let be the set of (conjugacy classes of) representations which are Schottky on every proper free factor. Hence by Lemma 3.2, and is still in the complement of . Let .
If then is dense in .
No point of can be in a domain of discontinuity for . Equivalently, any open invariant set in on which acts properly discontinuously must be contained in .
For (1), let and let be a proper free factor such that is not Schottky. We may assume has rank at least 2 since . Hence (as in the proof of Lemma 3.2) is approximated by representations with dense image. It follows that itself is approximated by representations dense on , so is dense in .
For (2), we will show that for every neighborhood of there is an infinite set of elements such that . Since is locally compact, this implies cannot be in any open set on which acts properly discontinuously.
Suppose first . Since is dense in , it suffices to consider the case that . Again let be a proper free factor on which is dense. We can assume that where is generated by one element . Now let be a sequence such that , and let be the automorphism that is the identity on and sends to . Note that has infinite order in . The number of powers of that take to any fixed neighborhood of itself goes to as , because . This concludes the proof for .
If , is empty. However, every has primitive element mapping to a non-loxodromic, so may be approximated by a representation sending a generator to an irrational elliptic. The same argument as above can then be applied to . ∎
We remark that is indeed strictly larger than , at least for even rank: If , represent the handlebody as an -bundle over a genus surface with one puncture, and let be a degenerate surface group with no accidental parabolics (i.e. every parabolic in is conjugate to the image of an element of , is discrete and faithful, and at least one end of the resulting manifold is geometrically infinite). The puncture cannot be in any proper free factor of , because it is represented by a curve on the boundary of the handlebody whose complement is incompressible. Hence the restriction of to any proper free factor has no parabolics. By the Thurston-Canary Covering Theorem  and the Tameness Theorem [1, 10], cannot be geometrically infinite, so it must be Schottky, and hence . On the other hand, every nonperipheral nonseparating simple curve on is primitive, and the ending lamination of a degenerate end can be approximated by such curves. Hence the uniform quasigeodesic condition fails on primitive elements, and cannot be in .
Is the interior of ?
In particular, in view of Lemma 5.1, a positive answer would imply
is the domain of discontinuity of acting on .
We remark that, a priori, there may not be any such domain, i.e. there may be no maximal set on which the action is properly discontinuous.
For the free group of rank 2, we have already seen that some of our statements are slightly different. In particular is empty since no one-generator subgroup of is dense. Moreover, is exactly the set for which every generator is loxodromic, and this is dense in since it is the complement of countably many proper algebraic sets.
Question 5.2 in particular asks, therefore, whether is dense. It is not clear (to the author) whether this is true, but there is some evidence against it (see below).
Another important feature of rank 2 is that the conjugacy class of the commutator of the generators, and its inverse, are permuted by automorphisms. It follows that the trace of the commutator is an -invariant function on , and one can therefore study level sets of this function.
The domain of discontinuity of was studied by Bowditch and Tan-Wong-Zhang . Bowditch defines the following condition on , which Tan-Wong-Zhang call condition BQ:
is loxodromic for all primitive .
The number of conjugacy classes of primitive elements such that is finite.
They show, using Bowditch’s work, that acts properly discontinuously on the invariant open set . It is still unclear whether is the largest such set.
Note that condition (1) is equivalent to membership in . It is evident that , and it seems plausible that they are equal.
Note also that computer experiments indicate that the intersection of with a level set of the commutator trace function is not dense in the level set (see Dumas ). In particular the slice corresponding to trace consists of type-preserving representations of the punctured-torus group, i.e. those with parabolic commutator, and empirically it seems that in this slice coincides with the quasifuchsian representations (which are all primitive-stable too, by Theorem 4.1 combined with Lemma 4.4 ). Bowditch has conjectured that this is in fact the case, and this seems to be a difficult problem. At any rate this appears to be evidence against the density of .
The question of the ergodic decomposition of on is still open. Note, in rank 2, the decomposition must occur along level sets of the commutator trace function. In rank 3 and higher our observations indicate that the simplest possible situation is that, outside , the action is ergodic, which we pose as a variation of Lubotzky’s original question:
Let . Is there a decomposition of into a domain where the action is properly discontinuous, and a set where it is ergodic? More pointedly, does act ergodically on the complement of in ?
In Gelander-Minsky  we show that in fact the action on is ergodic and topologically minimal. So if for example and have measure 0, we would have a positive answer for the above question.
It would also be nice to have a clearer understanding of the boundary of , and of which discrete representations contains.
From Lemma 3.2 we know that any discrete faithful representation with cusp curves that have compressible complement cannot be in . We’ve also mentioned the degenerate surface groups which are in but not .
If, however, is discrete and faithful without parabolics and is not Schottky, then it has an ending lamination which must lie in the Masur domain, and hence is blocking by Lemma 4.6. Hence it would be plausible to expect:
Every discrete faithful representation of without parabolics is primitive-stable.
More generally, a discrete faithful representation has a possibly disconnected ending lamination, whose closed curve components are parabolics. All the examples we have considered suggest this conjecture:
A discrete faithful representation of is primitive-stable if and only if every component of its ending lamination is blocking.
It might also be interesting to think about which representations with discrete image (but not necessarily faithful) are primitive-stable. In  we construct many primitive-stable representations whose images uniformize knot complements. What properties of a marked 3-manifold correspond to primitive stability?
Another interesting question is:
How do we produce computer pictures of ?
For rank , the character variety has complex dimension 2, and one can try to draw slices of dimension 1. Komori-Sugawa-Wada-Yamashita developed a program for drawing Bers slices, which are parts of the discrete faithful locus [31, 30], and Dumas refined this using Bowditch’s work . In particular what Dumas’ program is really doing is drawing slices of Bowditch’s domain BQ. If indeed , then this produces images of as well.
Other target groups
The discussion can be extended to other noncompact Lie groups, with moderate success. Let us consider first the case of for all , where is the case we have been considering. The definition of is unchanged, and stability of quasigeodesics works in all dimensions in the same way. Lemmas 2.1 Theorem 3.3 still hold. However, Sullivan’s theorem (Theorem 2.2) equating Schottky representations with those in the interior of is no longer available. Schottky representations in higher dimensions can be replaced by convex-cocompact representations: discrete and faithful, with convex hull of the limit set having a compact quotient. Now the conclusions of Lemma 3.2 must be changed somewhat: A convex-cocompact representation is certainly still primitive-stable, but it is not clear that For a primitive-stable , the proof of Lemma 3.2 shows that restricted to each proper free factor is in , but not that it is convex-cocompact.
For , the natural embedding of in clearly preserves primitive-stability and non-discreteness, so it is still true that contains indiscrete representations in higher dimension (and hence dense ones, by Lemma 2.1).
The case of is slightly trickier. When is even, we have given an example of a blocking curve that is the boundary of a one-holed surface, and so Theorem 4.1 shows that a Fuchsian structure on this surface, which gives an element of is primitive-stable but not Schottky. However when is odd we have no such example, and it is unclear to me if contains indiscrete elements. For and Goldman  has described the domain of discontinuity for the trace slice, and proved ergodicity in its complement.
For other noncompact rank-1 semisimple Lie groups, namely isometry groups of the non-homogeneous negatively curved symmetric spaces, primitive stability can again be defined in the same way. The hyperbolic plane always embeds geodesically in such a space, with its full isometry group acting. This is not easily extracted from the literature but is well-known; see Mostow , Bridson-Haefliger [8, Chap. II.10] and Allcock  for the requisite machinery, or note that these spaces are just the real, complex, and quaternionic hyperbolic spaces, and the Cayley plane, in all of which one can restrict to a real subspace. Thus, in these cases our examples for can be used.
Other cases, such as higher rank semisimple groups, presumably require a rethinking of the definitions, but there is again no reason to think that the geometric decomposition of Lemma 2.1 should be the right dynamical decomposition for the action of .
A completely different picture holds in the setting of non locally connected groups. In the case of , with a non-Archimedean local field of characteristic , as well as for a tree , Glasner  has shown that acts ergodically on .
Other domain groups
If we replace by for a closed surface , is replaced by , and the primitive elements are replaced by their natural analogue, the simple curves in the surface. The Schottky representations are replaced by the quasi-Fuchsian representations . We can define in a similar way, but now there is no good reason to think that is strictly larger than the quasi-Fuschsian locus. Indeed, every boundary point of can be shown not to lie in (nor in any domain of discontinuity for – see also Souto-Storm  for a related result), and this is because all parabolics are simple curves, and all ending laminations are limits of simple curves. It is still open as far as I know whether in fact is equal to ; this is closely related to (but formally weaker than) Bowditch’s conjecture in this setting.
One can also consider where is any fundamental group of a hyperbolic 3-manifold, and the dynamics are by ; see Canary-Storm . In general, the more complicated the group, the weaker we should expect the correspondence between the geometric and dynamical decompositions. An extreme example is when is a non-uniform lattice in . In this case has positive dimension, while Mostow rigidity tells us that is a single point. On the other hand Mostow also tells us that is finite in this case, so that it acts properly discontinuously on the whole of .
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