aa r X i v : . [ m a t h . G R ] D ec GROWTH SERIES OF CAT(0) CUBICAL COMPLEXES
BORIS OKUN AND RICHARD SCOTT
Abstract.
Let X be a CAT(0) cubical complex. The growthseries of X at x is G x ( t ) = P y ∈ V ert ( X ) t d ( x,y ) , where d ( x, y ) de-notes ℓ -distance between x and y . If X is cocompact, then G x is a rational function of t . In the case when X is the Daviscomplex of a right-angled Coxeter group it is a well-known that G x ( t ) = 1 /f L ( − t/ (1 + t )), where f L denotes the f -polynomial ofthe link L of a vertex of X . We obtain a similar formula for gen-eral cocompact X . We also obtain a simple relation between thegrowth series of individual orbits and the f -polynomials of variouslinks. In particular, we get a simple proof of reciprocity of theseseries ( G x ( t ) = ± G x ( t − )) for an Eulerian manifold X . Let X be a CAT(0) cube complex with a cocompact cellular actionby a group G . Denote by d ( x, y ) the ℓ -distance between vertices x and y of X . We consider the following growth series: G xy = X z ∈ Gy t d ( x,z ) — the growth series of G -orbit of y as seen from x , and G x = X y ∈ X t d ( x,y ) — the full growth series of X as seen from x .The aim of this paper is to establish relations between these growthseries and the local structure of X and X/G . In order to do this weintroduce more notation. The f -polynomial of a simplicial complex L is given by: f L ( t ) = X σ ∈ L t dim σ +1 . Note that we assume that L contains an empty simplex of dimension −
1, so the f -polynomial always has free term 1. For vertices x and y of X , denote by (cid:3) xy the cube spanned by x and y . In other words (cid:3) xy isthe minimal cube containing x and y . Let f xy denote the f -polynomialof the link of the cube (cid:3) xy , and let f x = f xx denote the f -polynomial Date : December 20, 2018.Both authors partially supported by a Simon’s Foundation Collaboration Grantfor Mathematicians. of the link of the vertex x . We put f xy = 0 if x and y are not containedin a cube.A fundamental example of a cocompact CAT(0) cube complex isthe Davis complex of a right-angled Coxeter group. In this case thegroup acts simply transitively on vertices and thus all the growth seriesare equal, and we have the following well-known result. (For generalCoxeter groups, this can be found in [6], Theorem 1.25 and Corollary1.29. For the right angled case, it takes the following form.) Theorem 1. If G is a right-angled Coxeter group and X is its Daviscomplex, then G x ( t ) f x (cid:18) − t t (cid:19) = 1 . In fact, it was proved by the second author in [3] that the sameformula holds if one assumes only that the f -polynomials of all verticesare the same: Theorem 2.
If the links of all vertices of X have the same f -polynomial,then G x ( t ) f x (cid:18) − t t (cid:19) = 1 . Our goal is to generalize this to the case of different links. Since, by aresult of Niblo and Reeves [2], CAT(0) cube groups have an automaticstructure, it follows that the growth series G xy are rational functionsof t computable in terms of the local structure of X . This computationwas carried out by the second author in [4], where it was used to provereciprocity of the growth series for Eulerian manifolds. In this paperwe obtain different and much simpler formulas for the growth serieswhich lead to an easy proof of reciprocity.Our result is easiest to state when the action is sufficiently free. De-fine:(1) c xy = (cid:18) − t − t (cid:19) d ( x,y ) f xy t − t ! . Theorem 3.
If the stars of vertices are embedded in
X/G , then thematrices ( G xy ) and ( c xy ) ( x, y ∈ X/G ) are inverses of each other.
Note that in this case the matrices are symmetric.In the general case, two vertices in
X/G can span multiple cubes. Toaccount for this we modify the coefficients c xy as follows. Let π : X → X/G denote the natural projection. For x, y ∈ X/G , pick ¯ x ∈ π − ( x ) and set ¯ c xy = X ¯ y ∈ π − ( y ) c ¯ x ¯ y . ROWTH SERIES OF CAT(0) CUBICAL COMPLEXES 3
Theorem 4.
The matrices ( G xy ) and (¯ c xy ) are inverses of each other: X y ∈ X/G ¯ c xy G yz = δ xz . Corollary 5. If X is an n -dimensional Eulerian manifold, then c xy and G xy satisfy reciprocity: c xy ( t − ) = ( − n c xy ( t ) ,G xy ( t − ) = ( − n G xy ( t ) . Proof.
For a simplicial Eulerian ( n − -sphere L we have the Dehn–Sommerville relations (see [5], pages 353-354 or [1], p.271) f L ( t −
1) = ( − n f L ( − t ) . A bit of algebra gives the first formula, and the second formula thenfollows. (cid:3)
At this point an attentive reader might wonder how to reconcile ourformula with the one for the Davis complex, where our matrices become × . We have the following lemma: Lemma 6. X y ∈ X c xy = f x ( − t t ) . X y ∈ X/G ¯ c xy = f x ( − t t ) . Proof.
The first statement is true for an n -cube as both sides evaluateto t ) n , and both sides behave the same under taking unions. Thesecond statement follows from the first. (cid:3) Summing the main formula P y ∈ X/G ¯ c xy G yz = δ xz over x , or z , orboth, and using the previous Lemma gives: Corollary 7. X y ∈ X/G f y (cid:18) − t t (cid:19) G xy = 1 , X y ∈ X/G ¯ c xy G y = 1 , X x ∈ X/G f x (cid:18) − t t (cid:19) G x = X/G.
BORIS OKUN AND RICHARD SCOTT
So we indeed recover the Davis complex formula.Our proof of Theorems 3 and 4 is based on a different description ofthe entries of the inverse of the matrix G xy . We develop this descriptionin the next four lemmas before proving the Theorems. For each vertex x ∈ X define a function h x : X → R [ t ] by h x ( y ) = t d ( x,y ) . The followinglemma is key in our approach. Lemma 8.
Let x ∈ X , and let S = V ert ( St ( x )) denote the verticesof the cubical star of x . Then the characteristic function of { x } , x isa unique linear combination of the functions h y , y ∈ S , over R ( t ) , thefield of rational functions.Proof. Since X is CAT(0), the hyperplanes near x (corresponding toedges starting at x ) divide X into convex polyhedral regions. Eachregion R has a unique vertex r closest to x . Also, r ∈ S . We will referto the regions as cones and to the vertices as cone points.For any z ∈ R and y ∈ S there exist a geodesic edge path which goesthrough r . d ( y, z ) = d ( y, r ) + d ( r, z ) . Therefore, h y ( z ) = t d ( r,z ) h y ( r ) . Rx zy r This implies that for a fixed cone the values of all the functions h y ( z ) are the same power of t multiple of the corresponding value at the conepoint. It follows that if a linear combination of h y , y ∈ S vanishes at acone point, then it vanishes on the whole cone.Thus, since the cone points are precisely S it is enough to prove thespecial case, when X = St ( x ) .In this case the S × S matrix of values of h -functions ( h y ( z )) = ( t d ( y,z ) ) has ’s on the diagonal and positive powers of t off the diagonal. Itsdeterminant is a nonzero polynomial, since it evaluates to at t = 0 . ROWTH SERIES OF CAT(0) CUBICAL COMPLEXES 5
Therefore, the matrix is invertible over R ( t ) and the desired coefficientsare given by the x -row of its inverse. (cid:3) Lemma 9. If X = A × B then the coefficients for X are products ofcoefficients for A and B .Proof. This is immediate from the formula h ( a,b ) ( x, y ) = h a ( x ) h b ( y ) . (cid:3) One of the implications of the proof of Lemma 8 is that it is enoughto understand the case when X = St ( x ) . Note that when X is a star,we do not need to assume that X is CAT(0).So assume that X = St ( x ) and denote the coefficients posited in theLemma by c Xxy . This should not cause confusion since we will show inLemma 11 that they are same as c xy given by (1). Our proof is basedon building X inductively cube by cube and using the product formulaand a certain inclusion–exclusion formula (Lemma 10 below.)In order to state the inclusion–exclusion formula we introduce morenotation. If A is a sub-complex of X containing x , which is also a star A = St A ( x ) , then we extend the coefficients c Axy to all of X by setting c Axy = 0 for y A . Since for y ∈ A the function h y for X restricts tothe function for A , we have X y ∈ X c Axy h y ( z ) = X y ∈ A c Axy h y ( z ) , and the resulting function restricts to x on A .The basis of our induction is the following. If X = { x } then theonly coefficient is 1. For a segment X = [ xy ] the coefficients are c Xxx =1 / (1 − t ) and c Xxy = − t/ (1 − t ) : x = 11 − t h x + − t − t h y . Lemma 10 (Inclusion–exclusion) . If X = St ( x ) decomposes as X = A ∪ C B , where A , B and C are subcomplexes of X which are also starsof x , then c Xxy = c Axy + c Bxy − c Cxy . Proof.
First consider a special case when A = C × [ xz ] is the star of theedge [ xz ] . We identify C with C ×{ x } . Then, for a = ( c, z ) ∈ A − C and b ∈ B we can choose a geodesic through c = ( c, x ) ∈ C and therefore h a ( b ) = th c ( b ) h b ( a ) = th b ( c ) . BORIS OKUN AND RICHARD SCOTT
Also, from the product formula we have c Axc = 11 − t c Cxc c Axa = − t − t c Cxc . It follows that X y ∈ A c Axy h y ( b ) = X a ∈ A − C c Axy h a ( b ) + X c ∈ C c Axy h c ( b )= X c ∈ C − t − t c Cxc th c ( b ) + X c ∈ C − t c Cxc h c ( b ) = X c ∈ C c Cxc h c ( b ) , X b ∈ B c Bxb h y ( a ) = t X b ∈ B c Bxb h b ( c ) = t x ( c ) . and X c ∈ C c Cxb h b ( a ) = t X b ∈ C c Cxb h b ( c ) = t x ( c ) . Therefore, X y ∈ X ( c Axy + c Bxy − c Cxy ) h y ( b ) = X y ∈ A c Axy h y ( b ) − X c ∈ C c Cxc h c ( b ) + X y ∈ B c Bxy h y ( b )= 1 x ( b ) , and X y ∈ X ( c Axy + c Bxy − c Cxy ) h y ( a ) = X y ∈ A c Axy h y ( a ) − X c ∈ C c Cxc h c ( a ) − X b ∈ B c Bxb h b ( a ) = 1 x ( a ) = 0 , since the bracketed differences vanish.Thus P y ∈ X ( c Axy + c Bxy − c Cxy ) h y = 1 x , and the special case follows fromuniqueness of the coefficients.The general case now follows by induction. Let X = A ∪ C B andlet [ xz ] be an edge in X . As before, associated to the edge we havedecomposition of X into the star of [ xz ] and the rest, which we writeas X = X ∪ X X , for which the inclusion-exclusion formula holds.(2) c Xxy = c X xy + c X xy − c X xy . Intersecting this decomposition with the original one gives decomposi-tions A = A ∪ A A , and similarly of B and C . It also gives decompo-sitions of X i = A i ∪ C i B i . The left hand sides of these 6 decompositions ROWTH SERIES OF CAT(0) CUBICAL COMPLEXES 7 are proper subsets of X and we can assume by induction that theinclusion–exclusion formula holds for them. c X xy = c A xy + c B xy − c C xy c X xy = c A xy + c B xy − c C xy c X xy = c A xy + c B xy − c C xy c Axy = c A xy + c A xy − c A xy c Bxy = c B xy + c B xy − c B xy c Cxy = c C xy + c C xy − c C xy Substituting the formulas in the first column into (2) and comparingwith c Axy + c Bxy − c Cxy using the second column verifies the desired formulafor X = A ∪ C B . (cid:3) Below are some examples. x − t − t − t x − t ) − t (1 − t ) t (1 − t ) − t (1 − t ) x − t ) − − t − t (1 − t ) t (1 − t ) − t (1 − t ) − − t − t − t (1 − t ) t (1 − t ) In fact, we have explicit formulas for the coefficients:
Lemma 11.
The coefficient of h y is precisely the c xy introduced before: c xy = (cid:18) − t − t (cid:19) d ( x,y ) f xy t − t ! . Proof.
By the product formula this is true if X is a cube, and bothsides behave the same under taking unions. (cid:3) We are now in position to finish the proof of the Theorems.
Proof.
Since c xy = 0 for x and y not spanning a cube, we have: X y ∈ X c xy h y = X y ∈ St ( x ) c xy h y = 1 x . BORIS OKUN AND RICHARD SCOTT
Since G yz = X w ∈ Gz h y ( w ) , we have X y ∈ X c xy G yz = X y ∈ Xw ∈ Gz c xy h y ( w ) = X w ∈ Gz x ( w ) = δ π ( x ) π ( z ) . Since G xy are G -invariant in both variables, and c xy is invariant underthe diagonal action, we can express this result in terms of X/G , toobtain Theorems 3 and 4. (cid:3)
Finally, we note that c xx = f x (cid:16) t − t (cid:17) . Taking t = √− we obtainthe following strange corollary: Corollary 12.
If the stars of vertices are embedded in
X/G , then tr ( c xy ( √− χ ( X/G ) . Thus the Euler characteristics can be computed from the matrix of thegrowth series ( G xy ) evaluated at √− . References [1] Ś. R. Gal. Real root conjecture fails for five- and higher-dimensional spheres.
Discrete Comput. Geom. , 34(2):269–284, 2005.[2] G. A. Niblo and L. D. Reeves. The geometry of cube complexes and the com-plexity of their fundamental groups.
Topology , 37(3):621–633, 1998.[3] R. Scott. Growth series for vertex-regular CAT(0) cube complexes.
Algebr.Geom. Topol. , 7:285–300, 2007.[4] R. Scott. Eulerian cube complexes and reciprocity.
Algebr. Geom. Topol. ,14(6):3533–3552, 2014.[5] R. P. Stanley.
Enumerative combinatorics. Vol. 1 , volume 49 of
Cambridge Stud-ies in Advanced Mathematics . Cambridge University Press, Cambridge, 1997.[6] R. Steinberg.
Endomorphisms of linear algebraic groups . Memoirs of the Ameri-can Mathematical Society, No. 80. American Mathematical Society, Providence,R.I., 1968.
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