R-trees and the Bieri-Nneumann-Strebel invariant

Publicacions Matemàtiques, Vol 38 (1994), 195-202 . R -TREES AN D THE BIERI--NEUMANN - STREBEL INVARIAN T GILBERT LEVIT T A bstract  Let G be a finitely generated group . We give a new characterization of its Bieri - Neumann - Strebel invariant E(G) , in terms of geometric abelian actions on R -trees . We provide a proof of Brown' s characterization of E(G) by exceptional abelian actions of G, usin g geometric methods . Introduction . In a 1987 paper at Inventiones [BNS], Bieri, Neumann and Strebe l associated an invariant E = E(G) to any finitely generated group G . This invariant may be viewed as a positively homogeneous open subse t of Hom(G, R) ~ {O} . It contains information about finitely generate d normal subgroups of G with abelian quotient . In the same issue of Inventiones [Bri, Brown introduced HNN-valua- tions and related E to actions of G on R -trees . In particular a nonzer o homomorphism x : G —> R is in E n —E if and only if R is the onl y R -tree admitting a minimal action of G with length function lxi (se e Theorem 3 .2 below) . A few months earlier, also in Inventiones [Le 1], this author studie d singular closed differential one-forms on closed manifolds M~ (ri > 3) . W e defined complete forms by several equivalent geometric conditions ; in the simplest case, a form w is complete if and only if every path in M is homotopic to a path - y that is either transverse to w or tangent to w (i .e . w (-y ' (t)} never vanishes or is identically O) . We proved thatany form cohomologous to a complete form is als o complete, so that completeness defines a subset U(M) in the De Rha m cohomology space 11-DR (M, R) Ham err1 M, R) . We also proved tha t U(M) depends only on the group G = Tr1 M, and in fact U(M) is nothin g but (7r i M) n —E(7r 1 M) . 196  G . LEVIT T In this note we use (a generalization of) closed one-forms to give a new characterization of (G), this time in terms of geometric actions o f G on R-trees (Theorem 3 .1) . Assuming for simplicity that G is finitel y presented, we say that an action of G on an R -tree is geometric if i t comes from a measured foliation on a finite 2-complex K with 7r 1 K= G (see [LP] for a complete discussion) . A consequence of Theorem 3 .1 is : Corollary . Let x : G - -} R be a nonzero homomorphism, with G finitely generated . (1) There exists a geometric action of G on an R -tree with length function ~ = Ixi if and only if x E EU —E . (2) The action of G on R by translations associated to x is geometri c if and only ifx E En —~ . We also give a geometric proof of Brown ' s theorem, by associating a natural R -tree T + (f) to any real -valued function defined on a pathconnected space (there is a similar construction in terms of romp -tree s in [BS, Chapter 11]) . In Part 1 we define closed one-forms relative to a homomorphism x : G ~ R, and we reformulate the condition x E E in terms of close d one-forms . In Part 2 we recali known facts about abelian actions o n R -trees (see [CuMo], [In Part 3 we prove both characterization s of E meñtioned aboye . Acknowledgements . This work was motivated by Skora's remar k that actions of nonabelian free groups on R are not geometric . It benefit - ted from conversations I had with R . Bieri during the Geometric method s in group theory semester in Barcelon a ' s Centre de Recerca Matemática , and in Frankfurt as part of the Procope program . 1 . Closed one-forms . Let x : G —> R be a homomorphism . A closed one-form relative t o x consists of a path-connected space X equipped with an action of G , together with a continuous function f : X ~ R such tha t f(9x) = f(x) + x( g ) forallxEXandgEG . The closed one-form is geometric if G acts as a group of coverin g transformations and the base X/G is (homeomorphic to) a finite CW - complex . Note that this forces G to be finitely generated . Example 1 . Let G be the trivial group . Then any function on a path-connected space defines a closed one-form . TREES AND BIERI-NEUMAN - STREBEL INVARIANT  19 7 Example 2 . Let co be a closed differential one-form on a closed manifold M . Let x : 7r 1 M -> R be the homomorphism given by integratin g w along loops . Let p : X — } M be the universal covering . Then an y f : X —> R such that df = p * cv defines a geometric closed one-for m relative to x . Example 3 . Let 1' be the Cayley graph of G relative to some fixe d generating system . Given a homomorphism x : G -- } R, view it as a func - tion on the set of vertices of T, and extend it affinely and G-equivariantl y to a function f defined on the whole of F . This defines a closed one-for m relative to x . It is geometric if and only if the generating system is finite . Example 4 . Any abelian action of G on an R -tree T defines a close d one-form f : T -> R (see Part 2) . If f is a closed one-form on X, we denote X > c = f '(c, +oo ) fo r cER . Proposition 1 .1 . Let f : X R be a geometric closed one-for m relative to a nonzero homomorphism x : G —> R . Far any c E R, th e set X > c has at least one component on which f is unbounded . Thi s component is unique if and only if x E E(G) . Proof : Since f is geometric, the group G acts on X as a group o f covering transformations . Let X ' = X/G', where G ' is the commutato r subgroup of G . The function f induces f : X ' --> R . Let  = (c, +oo) . By [BNS, Part 5], there exists a unique component A ' o f on which f ' is unbounded, and x E E(G) if and only if the natura l map from 7r 1 A ' to G ' is onto (compare [Le 1, Parts IV and V] and [Sil) . The proposition follows . ■ 2 . Abelian actions on R -trees . Suppose a finitely generated group G acts by isometries oil an R -tre e T . The length function .e : G - -> R + is defined as « g} = inG E T d(x, gx) . The action is trivial if there is a global fixed point (equivalently if ~ 1-7 . O) , minimal if there is no proper invariant subtree . The action (or th e length function) is called abelian if ,e is the absolute value of a nonzer o homomorphism x : G —+ R . Two minimal actions of G with the sam e length function £ are equivariantly isometric, except maybe if .e is abelian . Brown's theorem (Theorem 3 .2 below) is concerned with this "maybe" . A nontrivial action is abelian if and only if there is a fixed end e . W e can then define a closed one-form on T, as follows . Given x E T, there i s a unique isometric embedding i x : (—oo, o] ---> T such that ix(—oo) = e 198  G . LEVIT T and ix (o) = x . Fixing a basepoint m E T, we define f (x) as the only rea l number such that i x (t) = im (t + f (x)) for Itl large enough ( " Buseman n function" ) . Then f is a closed one-form on T, relative to some nonzer o homomorphism x : G -> R satisfying ~ = lxl . This homomorphis m measures how much elements of G push away from e . An abelian action is called exceptional if there is only one fixed en d e . We can then define x unambiguously, and we say that the action i s associated to x . If there are two fixed ends ( i .e . if there is an invarian t line), we say that the action is associated to both x and —x . 3 . Characterizations of E . Let f : X — } R be continuous, with X path-connected . Assume f ha s bounded variation in the following sense : given x, y E X , there exists a pathy : [0,1] --} X from x to y such that f o y has bounded variation . The infimum of the total variation of f o y over all paths y from x to y then defines a pseudometric d(x, y) on X . We let T( f) be the associated metric space : points of T( f) are equivalence classes for the relation d (x, y) = O . Denote 7r : X --> T( f) th e natural pro j ect ion and a : T( f) —> R the map such that a o 7r = f . If f is a closed one-form relative to some x : G —+ R, there is a n induced isometric action of G on T( f) with A( gx ) = a(x) + x(g) . Whe n T( f) is an R -tree, the length function .~ of this action satisfies .e ~ ix I (since a does not increase distances) . Definition . Consider an abelian action of a finitely generated grou p G on an R -tree T, associated to x : G —> R . The action is geometric i f and only if there exists a geometric closed one-form f : X ~ R relativ e to x such that T( f) is G -equivariantly isometric to T . Theorem 3 .1 . Let x : G — } R be a nontrivial homornorphism, wit h G a finitely generated group . There exists a geometric abelian action o f G on an R -tree associated to x if and only ifx E —~ . Pro o f Let f : X - -} R be a geometric closed one-form relative to x . Assum e that T( f) is an R -tree and the action of G on T( f) is abelian, associate d to x . We show x E —E . Fix g E G with x(g) C 0, and fix x E X with, say, f(x) = O . For A large enough, the path component U of f — l (—A, A) containing x meet s every orbit for the action of G on X : this is because X/G is a finit e complex . We may also assume that A has been chosen so that gx E U . TREES AND BIERI-NEUMAN - STREBEL INVARIANT  19 9 We then claim that any y E X with f(y) Ç —A belongs to the sam e component of f -1 (— oo, A) as x . This will imply x E —E by Proposition 1 .1 . Choose an infinite path y : [0, +oo) --> X such that is a pat h from x to gx in Uand y(t + n) = g n y(t) for n E N and t E [0, 11 . Sinc e x(g) C 0 this path is contained in f -1 (—Qo, A) . Given y E X with f(y) Ç —A, fix h E G such that hy E U, and choose a path 6 from hy to x in U . Consider the infinite path p obtaine d by applying h —1 to Sy : it starts at y and passes through h —1 x, h — 1 gx , h'g 2 x, . . . . It is contained in f'(—oo, A) since f (y) ç —A . The image of y in T( f) contains all points gn 7r 1xl (n E N), while th e image of p contains a11 points h —l g n 7r(x) _ (h 1 gh)ir(h'x) . Now th e translation axes of g and hT 1 gh intersect in a half-line containing th e fixed end e (unless they are equal). Furthermore g and h'gh both pus h towards e since x(g) C O . It follows that y and p are contained in th e same component of f -1 ( — oo, A), so that x E E . Conversely, supposex E —~ . First assume that G is finitely presented . Let M be a closed manifold with 7r1 M = G . Consider a geometric close d one-form f : X —} R as in Example 2 of Part 1 . To fix ideas we ma y assume that f is a Morse function . Since X is simply connected (it is the universal covering of M), it i s known [GS] that T(f) is an R -tree (see [Le 2, Corollary 111 .5]) . Sinc e x E —E the function A : T( f ) R is bounded on every component o f a —1 (--oo, c) but one . It foliows that the action of G on T( f) is abelian , associated to x : letting a go to —oo defines an end e of TU) which i s invariant under the action . Now let G be any finitely generated group . Using (i) <=> (iii) in [BNS , Proposition 2 .1] we can find an epimorphism q : H --> G, with H finitel y presented, such that x ' ~ x o q belongs to —E(H) . Apply the previou s construction to H and x ' . Let Y be the normal covering of M wit h group G and g : Y --> R the map induced by f . The length function of the action of H on the R -tree T( f) is .e == j It vanishes on the kernel K of q . It follows from [Le 2, corollary o f Theorem 2] that T(g) = T( f)/K is an R -tree . The action of G on thi s R-tree is abelian, associated to x . ■ Theorem 3 .2 (Brown) . Let x : G --> R be a nontrivial homomorphism, with G finitely generated . Then x E E if and only if there exist s no exceptional abelian action associated to x . We start the proof with a general construction . Let f : X —> R b e continuous, with X path connected . We construct an R -tree T + (f) as 200  G . LEVIT T follows . Given x, y E X, defin e S(x, y) = f (x) + f (y) - 2 sup min f (ry(t)) , ry tE[0,1 ] the supremum being over all paths r y : [0,1] -~ X from x to y . This is a pseudodistance on X and we let T + (f) be the associated metric space . Proposition 3 .3 . The space T + ( f) is an R -tree . If p, : T + ( f) --> R is the map induced by f, all sets p, -1 (-oo, c) are path-connected, so tha t T + ( f) has a preferred end e = p,---1(---00) . Proof : We first prove that T + (f) is an R -tree . By [AB, Theorem 3 .17] i t suffices to show that S satisfies the 0-hyperbolicity inequalit y s(x, y) + s(z, t) < m ax{s(x, z) + s( y , t), s(x, t) + s(y, z)} . By linearity we need only worry about the terms S ' = sup min f ory . The y satisfy inequalities such a s 6' (x, y)  min {max(S ' (x, z), 6' (z, y)), max(S ' (x, t), 6' (t, y)) } and we conclude by applying the inequalit y min {max(a, c), max(b, d)} +min {max(a, b), max(c, d)} < max(a+d, b+c) , valid far any four real numbers a, b, c, d . Let 7r + be the projection from X to T + ( f ) . Given x, y E X i n f --1 (--oo, c) with, say, f(y) ç f(x), choosea path y : [0,1] —> X fro m x to y . If (p, g) is a maximal interval in (fo'y)'(f(x),+oo), we hav e (7r + o y)( p) = (7r+ o ¡ y )(g) and we can change 7r + o y on (p, q) so tha t it becomes constant on [p, q] . Doing this for all intervals (p, q) yields a path from 7r + (x) to 7r + (y) in 11 -1 (--00, c) . ■ If f is a closed one-form relative to x : G -4 R, the natural action o f G on T + (f) fixes e . It is abelian, associated to x (note that this actio n is nongeometric whenever x 1 -~, by Theorem 3 .1) . To prove Theorem 3 .2, wefix a finite generating system S for G wit h x(s) > 0 for every s E S and we consider the corresponding Cayley grap h F . First assume x « E . Let f : F --> R be as in Example 3 of Part 1 . W e claim that the abelian action of G on T + (f) is exceptional . TREES AND BIERI-NEUMAN - STREBEL INVARIANT  20 1 Let u 1 and u2 be vertices of I' belonging to distinct components U 1 , U 2 of some f -1 (c, +oo) . Fix s E S . The whole ray '2 .Gi, u i s, uis 2 , . . . , u 2 s n , . . . is contained in U i . Writing ui s n = ( usu 1 )u i we see that u 1 su1 1 an d u2 su2 1 do not have the same translation axis in T + ( f ) . This means tha t the action is exceptional . Now assume x E E . Let T be an R -tree with a minimal abelian actio n associated to x . We show that the action is not exceptional . Choose x E T belonging to the translation axis of every s E S . Consider a G -equivariant map (p : F --~ T, affine on each edge, sending 1 t o x . It is surjective because the action is minimal . Define f : T —> R as in Part 2 . The choice of x implies that g = f o (p i s monotonous on each edge of F . It follows that g is unbounded on ever y component of g —1 (c, +oo) , so that g —1 (c, +oo) is connected for every c E R by Proposition 1 .1 . Projecting to T we see that every f —1 (c, +oo ) is connected : the action is not exceptional . ■ Combining Theorems 3 .1 and 3 .2 we get : Corollary . Let x : G—> R be a nonzero homomorphism, with G finitely generated . (1) If x E En --- ~, the action of G on R by translations is the onl y minimal action with length function .Q = IxL It is geometric . (2) If x E E but x « —E, there exist geometric exceptional abelian actions associated to —x . The only minimal action associated t o x is the action on R, it is not geometric . (3) If x « E U —E, there exist both exceptional abelian actions associated to x and exceptional actions associated to —x . No actio n with length function  is geometric . C ombining wit h Theorem B .1 from [B NS] we obtain : Corollary . Let G be finitely generated . The following conditions ar e equivalent : (1) Every nontrivial action of G on R by translatiorisis geometric . (2) The commutator subgroup G ' is finitely generated . Reference s [AB] R . ALPERIN, H . BASS, Length functions of group actions o n A -trees, in " Combinatorial group theory and topology," S . M . Gersten, J . R . Stallings, ed ., Ann . Math . Studies 111, Princeton Univ . Press, 1987 . 202  G . LEVIT T [BNS] R . BIERI, W . D . NEUMANN, R . STREBEL, A geometric invari - ant of discrete groups, Invent . Math . 90 (1987), 451-477 . [BS] R . BIERI, R . STRÈBEL, " Geornetric invariants for discret e groups, " W . de Gruyter, Monograph to appear . [Br] K . S . BROWN, Trees, valuations, and the Bieri - Neumann - Strebe l invariant, Invent . Math . 90 (1987), 479-504 . [CuMo] M . CULLER, J . W . MORGAN, Group actions on R -trees, Proc . 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