A Generalized Axis Theorem for Cube Complexes
Abstract.
We consider a finitely generated virtually abelian group acting properly and without inversions on a CAT(0) cube complex . We prove that stabilizes a finite dimensional CAT(0) subcomplex that is isometrically embedded in the combinatorial metric. Moreover, we show that is a product of finitely many quasilines. The result represents a higher dimensional generalization of Haglund’s axis theorem.
1. Introduction
A CAT(0) cube complex is a cell complex that satisfies two properties: it is a geodesic metric space satisfying the CAT(0) comparison triangle condition, and each cell is isometric to . We will call this metric the CAT(0) metric and refer to [2] for a comprehensive account. A hyperplane is the subset of points equidistant between two adjacent vertices. Despite the brevity of this definition, hyperplanes are better understood via their combinatorial definition, and the reader is urged to consult the literature; see [10] [7] [12] for the required background. There also exists an alternative metric on the cubes of , that we will refer to as the combinatorial metric , sometimes referred to as the metric. The combinatorial distance between two cubes is the length of the shortest combinatorial path in joining the cubes. Equivalently, the combinatorial distance between two cubes is the number of hyperplanes in separating them. We will always assume that a group acting on a CAT(0) cube complex preserves its cell structure and maps cubes isometrically to cubes. A group acts without inversions if the stabilizer of a hyperplane also stabilizes each complementary component. The requirement that the action be without inversions is not a serious restriction as acts without inversions on the cubical subdivision.
A connected CAT(0) cube complex is a quasiline if it is quasiisometric to . The rank of a virtually abelian group commensurable to is . The goal of this paper will be the following theorem:
Theorem 4.3.
Let be virtually . Suppose acts properly and without inversions on a CAT(0) cube complex . Then stabilizes a finite dimensional subcomplex that is isometrically embedded in the combinatorial metric, and , where each is a cubical quasiline and . Moreover, is a codimension1 subgroup for each hyperplane in .
Note that will not in general be a convex subcomplex.
Corollary 1.1.
Let be a finitely generated virtually abelian group acting properly on a CAT(0) cube complex . Then acts metrically properly on .
Corollary 1.2.
Let be a finitely generated group acting properly on a CAT(0) cube complex . Then virtually subgroups are undistorted in .
Let be an isometry of , and let . The displacement of at , denoted , is the distance . The translation length of , denoted , is . Similarly, if is a cube of , we can define the combinatorial displacement of at , denoted , as and the combinatorial translation length, denoted , is . Note that , and are conjugacy invariant. An isometry of a CAT(0) space is semisimple if for some , and acts semisimply on a CAT(0) space if each is semisimple.
If a virtually group acts metrically properly by semisimple isometries on a CAT(0) space , then the Flat Torus Theorem [2] provides a invariant, convex, flat . A group acting on a CAT(0) cube complex does not, in general, have to do so semisimply. See [1] for examples of nonsemisimple isometries in Thompson’s group acting on an infinite dimensional CAT(0) cube complex. Alternatively, in [6] a freebycyclic group is shown not to permit a semisimple action on a CAT(0) space. Yet in [13] it is shown that does act freely on a CAT(0) cube complex. Thus Theorem 4.3 can be applied to such actions, whereas the classical Flat Torus Theorem cannot.
A virtually abelian subgroup is highest if it is not virtually contained in a higher rank abelian subgroup. If is a highest virtually abelian subgroup of a group acting properly and cocompactly on a CAT(0) cube complex , then cocompactly stabilizes a convex subcomplex which is a product of quasilines, as above [14]. However, this theorem fails without the highest hypothesis. Moreover, most actions do not arise in the above fashion.
Despite the fact that the Flat Torus Theorem will not hold under the hypotheses of Theorem 4.3, we can deduce the following:
Corollary 4.4.
Let be virtually . Suppose acts properly and without inversions on a CAT(0) cube complex . Then cocompactly stabilizes a subspace homeomorphic to such that for each hyperplane , the intersection is either empty or homeomorphic to .
The initial motivation for Theorem 4.3 and Corollary 4.4 was to resolve the following question posed by Wise. Although we have not found a combinatorial flat, Corollary 4.4 is perhaps better suited to applications (see [15]).
Problem 1.3.
Let act freely on a CAT(0) cube complex . Does there exists a equivariant map where is a square complex homeomorphic to , and such that no two hyperplanes of map to the same hyperplane in ?
A combinatorial geodesic axis for is a invariant, isometrically embedded, subcomplex with . Note that realizes the minimal combinatorial translation length of . Theorem 4.3 is a high dimensional generalization of Haglund’s combinatorial geodesic axis theorem. Haglund’s proof involved an argument by contradiction, exploiting the geometry of hyperplanes. We reprove the result in Section 5 by using the dual cube complex construction of Sageev. The results are further support for Haglund’s slogan “in CAT(0) cube complexes the combinatorial geometry is as nice as the CAT(0) geometry”.
The following is an application of Theorem 4.3, and the argument is inspired by the solvable subgroup theorem [2]. Note that since we do not require that the action of on a CAT(0) cube complex be semisimple the following is not covered by the solvable subgroup theorem.
Corollary 1.4.
Let be virtually , and let be an injection with for all . Then cannot act properly on a CAT(0) cube complex.
Proof.
Suppose that acts properly on a CAT(0) cube complex . After subdividing we can assume that acts without inversions. As is finitely generated, there exists an in the finite generating set such that for all , otherwise for some , contradicting our hypothesis. Thus, . By Theorem 4.3 there is an equivariant isometrically embedded subcomplex such that where each is a cubical quasiline.
As is isometrically embedded in in the combinatorial metric, the combinatorial translation length is the same in as it is in . The set must be unbounded since the action of on is proper and is locally finite. However, since is conjugacy invariant in , we conclude that for all . Thus, we arrive at the contradiction that is both bounded and unbounded. ∎
However, we have the following example of a solvable group which does act freely on a CAT(0) cube complex.
Example 1.5.
Let . Note that is the fundamental group of the nonpositively curved cube complex obtained from a cube , and cubes with cubes inserted for every cardinality collection of cubes to create an torus. One should think of as an infinite cubical torus. The oriented loop represents the element .
Let be the monomorphism such that . Let be the associated ascending HNN extension. Note that is generated by and . There is a graph of spaces obtained by letting be the vertex space and be the edge space and identifying and with , and the cube with and with . Note that is nonpositively curved, and therefore acts freely on the CAT(0) cube complex , the universal cover of .
Acknowledgements: I would like to thank Daniel T. Wise, Mark F Hagen, Jack Button, Piotr Przytycki, and Dan Guralnik.
2. Dual Cube Complexes
Let be a set. A wall in is a partition of into two disjoint, nonempty subsets. The subsets are the halfspaces of . A wall separates if they belong to distinct halfspaces of . Let . A wall intersects if nontrivially intersects both and . Let be a set of walls in , then is a wallspace if for all , the number of walls separating and is finite. If intersects , then the restriction of to , is the wall in determined by .
In this paper duplicate walls are not permitted in . Let be the set of halfspaces of corresponding to .
Example 2.1.
Let be a CAT(0) cube complex, and let be a hyperplane in . The complement has two components, therefore defining a wall in such that is an open halfspace not containing , and is a closed halfspace containing . Note that . Let and denote the maximal subcomplexes contained in and respectively. Note that and are convex subcomplexes. Let be the set of walls determined by the hyperplanes in . Then is the wallspace associated to . Note that we are using to denote both the hyperplane and the wall corresponding to the hyperplane.
A function is a cube if and the following two conditions are satisfied:

For all the intersection is nonempty.

For all , the set is finite.
The dual cube complex is the connected CAT(0) cube complex obtained by letting the union of all cubes be the skeleton. Two cubes are endpoints of a cube if for all but precisely one . An cube is then inserted wherever there is the skeleton of an cube. The hyperplanes in are identified naturally with the walls in . A proof of the fact that is in fact a CAT(0) cube complex can be found in [9].
A point determines a cube defined such that for all . Condition (1) holds immediately since for all . Condition (2) holds for , since if a wall does not separate and , we can deduce that , hence all but finitely many satisfy . Such cubes are called the canonical cubes.
Lemma 2.2.
Let be a CAT(0) cube complex. Let be a set of walls obtained from the hyperplanes in . Let be a connected subcomplex of , and let be the subset of walls intersecting . Let be walls in restricted to . Then is a wallspace and embeds in isometrically in the combinatorial metric.
Proof.
We first claim that the map is an injection. Suppose that are distinct walls. As intersects , and since is connected, there are cubes in that are dual to the hyperplanes corresponding to . Therefore, both cubes in belong in a single halfspace of , so .
We construct a map on the skeleton first. Let be a cube in . We let be the uniquely defined cube such that for , and for . To verify that is a cube, first observe that is nonempty since . Secondly, if we need to show that for all but finitely many . Choose , then for all , hence for all . Let be the set of walls in separating and . Then for all .
The cubes are embedded since if , there exists such that , hence . If are adjacent cubes in , then for all , with the exception of precisely one wall . Therefore, we can deduce that for all walls in , with the precise exception of . Therefore, the skeleton of embeds in , which is sufficient for to extend to an embedding of the entire cube complex.
Consider as a subcomplex of . The set of hyperplanes in embeds into the set of hyperplanes in . To see that is an isometrically embedded subcomplex, let be cubes in and be a geodesic combinatorial path in joining them. Each hyperplane dual to in intersects precisely once, and since the hyperplanes in inject to hyperplanes in , it is geodesic there as well. ∎
Given a wall associated to a hyperplane in we let denote the carrier of , by which we mean the union of all cubes intersected by .
The following Lemma decribes what is called the restriction quotient in [3].
Lemma 2.3.
Let be a set and let be a set of walls of . Let be a group acting on . Let be a invariant subset. Then there is a equivariant function . Moreover, is nonempty for all cubes in .
Proof.
Let be a cube in . Let for . It is immediate that is equivariant.
To verify is a cube in first note that for all , since for all . Secondly, for all observe that for all but finitely many . Indeed, this is true for all but finitely many .
To see that is nonempty for all cubes in we determine a cube in such that . Fix . Let for . Suppose that . If for some let . Similarly if . Otherwise, if intersects for all then let .
To verify that is a cube, consider the following cases to show for . If then . Suppose that and for some . If , then . If and for some then . If intersects for all , then . Finally if both and , then their intersection will contain at least .
Finally, we verify that for there are only finitely many such that . Suppose, by way of contradiction, that there is an infinite subset of walls such that for all . We can assume, by excluding at most finitely many walls, that each . Similarly, by excluding finitely many walls, we can assume that does not separate and . Therefore, for . Therefore, by construction of , there exist such that , which implies that . There are infinitely many distinct , as otherwise there is a such that for infinitely many , which would imply that infinitely many separate from an element in the complement of . Therefore, infinitely many distinct walls have , contradicting that is a cube in . ∎
3. Minimal invariant convex subcomplexes
The following is Theorem 2 from [5]. As this paper is written in Russian, we give a proof in Appendix A based on the work in [8] as well as stating the definition of codimension1.
Theorem 3.1 (Gerasimov [5]).
Let be a finitely generated group that acts on a CAT(0) cube complex without a fixed point or inversions. Then there is a hyperplane in that is stabilized by a codimension1 subgroup of .
The goal of this section is to prove the following:
Lemma 3.2.
Let be a finitely generated group acting without fixed point or inversions on a CAT(0) cube complex . There exists a minimal, invariant, convex subcomplex such that contains only finitely many hyperplane orbits, and every hyperplane stabilizer is a codimension1 subgroup of .
Proof.
Since is finitely generated, by taking the convex hull of a orbit we obtain a invariant convex subcomplex containing finitely many orbits of hyperplanes. Assume that is a minimal such subcomplex in terms of the number of hyperplane orbits.
Let be the wallspace obtained from the hyperplanes in . Suppose that is not a codimension1 subgroup of for some . Let be the orbit of . By Lemma 2.3 there is an invariant map . Since is not commensurable to a codimension1 subgroup, Theorem 3.1 implies that there is a fixed cube in . Lemma 2.3 then implies that is nonempty. Assuming that , then the intersection contains a proper, convex, invariant subcomplex of , with one less hyperplane orbit. This contradicts the minimality of . ∎
The following Corollary follows since all codimension1 subgroups of a rank virtually abelian group are of rank .
Corollary 3.3.
Let be a rank , virtually abelian group acting without fixed point or inversions on a CAT(0) cube complex . Then there exists a minimal, invariant, convex subcomplex such that contains only finitely many hyperplane orbits, and every hyperplane stabilizer is a rank subgroup of .
4. Proof of Main Theorem
Definition 4.1.
Regard as a CAT(0) cube complex whose skeleton is . Let be an isometry of . A geodesic combinatorial axis for is a invariant subcomplex homeomorphic to that embeds isometrically in .
Definition 4.2.
Let be a metric space. The subspaces are coarsely equivalent if each lies in an neighbourhood of the other for some .
Theorem 4.3.
Let be virtually . Suppose acts properly and without inversions on a CAT(0) cube complex . Then stabilizes a finite dimensional subcomplex that is isometrically embedded in the combinatorial metric, and , where each is a cubical quasiline and . Moreover, is a codimension1 subgroup for each hyperplane in .
Proof.
By Corollary 3.3 there is a minimal, nonempty, convex subcomplex stabilized by , containing finitely many hyperplane orbits, and is a rank subgroup of , for each hyperplane .
Let be a generating set for . Let be a cube. Let be the Cayley graph of with respect to . Let be a equivariant map that sends vertices to vertices, and edges to combinatorial paths or vertices in . Let . As acts properly on , and cocompactly on , the graph is quasiisometric to . Let be the set of hyperplanes intersecting , and let be the associated wallspace. By Lemma 2.2 we know that is an isometrically embedded subcomplex of . Fix a proper action of on , and let be a equivariant quasiisometry. Note that is a quasiisometrically embedded subgroup of , for all . Thus is coarsely equivalent to a codimension1 affine subspace . Moreover, and are coarsely equivalent to the halfspaces of .
Let . Since there are finitely many orbits of hyperplanes in , there are only finitely many commensurability classes of stabilizers. Therefore, we may partition as the disjoint union where each contains all walls with commensurable stabilizers. For each let be coarsely equivalent to a codimension1 affine subspace , stabilized by . If then and are nonparallel affine subspaces, and therefore and will intersect in . Therefore, every wall in intersects every wall in if , and thus .
Finally, we show that is a quasiline for each . As permutes the factors in , there is a finite index subgroup that preserves each factor. For each , the stabilizers are commensurable for all . Therefore, there is a cyclic subgroup that is not virtually contained in any and thus acts freely on . As the stabilizers of are commensurable, all will be quasiequivalent to parallel codimension1 affine subspaces of , which implies that only finitely many translates of can pairwise intersect. As there are finitely many orbits of in , there is an upper bound on the number of pairwise intersecting hyperplanes in . Thus, there are finitely many orbits of maximal cubes in , which implies that is CAT(0) cube complex quasiisometric to . ∎
We can now prove Corollary 4.4.
Corollary 4.4.
Let be virtually . Suppose acts properly and without inversions on a CAT(0) cube complex . Then cocompactly stabilizes a subspace homeomorphic to such that for each hyperplane , the intersection is either empty or homeomorphic to .
Proof.
By Theorem 4.3 there is a equivariant, isometrically embedded, subcomplex , such that , where each is a quasiline, and is a codimension1 subgroup. Considering with the CAT(0) metric, note that is a complete CAT(0) metric space in its own right, and acts semisimply on . By the Flat Torus Theorem [2] there is an isometrically embedded flat . Note that is not isometrically embedded. As is a codimension1 subgroup of for each hyperplane in , the intersection is either empty or, as is isometrically embedded, the hyperplane intersection is an isometrically embedded copy of . ∎
5. Haglund’s Axis
The goal of this section is to reprove the following result of Haglund as a consequence of Corollary 4.4.
Theorem 5.1 (Haglund [7]).
Let be a group acting on a CAT(0) cube complex without inversions. Every element either fixes a cube of , or stabilizes a combinatorial geodesic axis.
Proof.
As finite groups don’t contain codimension1 subgroups, Theorem 3.1 implies that if is finite order then it fixes a cube. Suppose that does not fix a cube, then must act properly on . By Corollary 4.4, there is a line stabilized by , that intersects each hyperplane at most once at a single point in . Let be the set of hyperplanes intersecting . Note that the intersection points of the walls in with is locally finite subset.
Fix a basepoint that doesn’t belong to a hyperplane intersecting , and let be the canonical cube corresponding to . Let be the set of hyperplanes separating and , and assume that . Reindex the hyperplanes such that . The ordering of the hyperplanes separating and determines a combinatorial geodesic joining and of length , where the th edge is a cube dual to . This can be extended equivariantly, to obtain a combinatorial geodesic axis , since each hyperplanes intersects at most once. ∎
Appendix A Codimension1 Subgroups
Definition A.1.
Let be a finitely generated group. Let denote the Cayley graph of with respect to some finite generating set. A subgroup is codimension1 if has more than one end.
Let denote the operation of symmetric difference. A subset is finite if where is some finite subset of . We will use the following equivalent formulation (see [11]) of codimension1: A subgroup is a codimension1 subgroup if there exists some such that

,

is almost invariant, that is to say that is finite for any .

is proper, that is to say that neither nor is finite.
Theorem A.2.
Let be a finitely generated group acting on a CAT(0) cube complex without edge inversions or fixing a cube. Then the stabilizer of some hyperplane in is a codimension1 subgroup of .
Proof.
Suppose that no hyperplane stabilizer is a codimension1 subgroup of . We will find a cube fixed by .
Let denote the set of hyperplanes in . We can assume that has finitely many orbits of hyperplanes after possibly passing to the convex hull of a single cube orbit in . If are cubes in , then let denote the hyperplanes separating and . Note that
Let be a minimal set of representatives of those orbits. Let
We can verify that satisfies the first two criteria in Definition A.1.
(1): It is immediate that , as doesn’t invert the hyperplanes in .
(2): Let xor denote the exclusive or. For we can deduce that is finite:
As is a wallspace, there are only finitely many such that separates and . If are the translates then
which implies almost invariant.
Therefore, cannot be proper, for any , as we have assumed that none of the are codimension1. This means that is finite so where is finite.
Claim.
for all .
Proof.
As the final set is covered by translates of , we can deduce that there are at most hyperplanes in . ∎
Thus, we can conclude that the orbit of is a bounded set. If has a finite orbit in , then the convex hull of the orbit is a compact, finite dimensional, complete CAT(0) cube complex, and we can apply Corollary II.2.8 (1) from [2] to find a fixed point . As is in the interior of some cube that is fixed by , and since doesn’t invert hyperplanes we can deduce that fixes a cube in that cube. If the orbits in are infinite, then their convex hull may not be complete, so the above argument will not hold.
Let denote the connected cube, a graph with vertices given by functions with finite support, and edges that join a pair of distinct vertices if and only if they differ on precisely one hyperplane.
Fix a cube . Then there is an embedding
that maps the cube to where
A hyperplan separates two vertices in if . Note that separates cubes in if and only if it separates and . Therefore, we can define for vertices in and conclude that if are cubes in then . This implies that is an isometric embedding in the combinatorial metric.
We will show that a bounded orbit in implies there is a fixed cube in and then argue that we can go one step further and find a fixed cube in .
Let be the Hilbert Space of square summable functions . There is an embedding given by
It is straight forward to verify that . There is a action on such that if then
It is again straight forward to verify that this action is by isometries, and that is equivariant.
As is bounded, so is . It then follows that has a fixed point in ([8] gives a proof and also cites Lemma 3.8 in [4]). Let be the fixed point. For all we can deduce that is either or . Therefore can only take two values on the hyperplanes in a single orbit. As has to be square summable the two values have to be and , and can only take the value on finitely many hyperplanes. Thus, is the image of a point in .
Let be a invariant vetex which minimizes the distance to the image of in . Let be a orbit of cubes in such that realize the minimal distance from .
Let be the set of hyperplanes that intersect . Every hyperplane in must intersect otherwise if is the finite, invariant subset of hyperplanes separating from we can define a cube such that
and deduce that is invariant and is closer to than .
Let an enumeration of cubes in . Each hyperplane separating and must lie in either or . As is minimal distance in from , the edges in incident to must be dual to hyperplanes not in , and instead belongs to . Therefore, the hyperplane dual to the first edge in a combinatorial geodesic joining to must lie in . Similarly, there exists a hyperplane dual to the first edge of the combinatorial geodesic in joining to that belongs to but not . Note that cannot intersect in , otherwise would be dual to an edge incident to , which would imply that there exists a cube in adjacent to that is closer to . Therefore separates from in . Iterating this argument produces a sequence of disjoint hyperplanes such that separates from in . This contradicts the hypothesis that is a bounded set in . ∎
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