COMPLETE TREES FOR GROUPS WITH A REAL-VALUED LENGTH FUNCTION ROGER C. ALPERIN AND KENNETH N. MOSS
Following Lyndon's axiomatic treatment of Nielsen's cancellation arguments for free products [5], I. M. Chiswell showed the equivalence of the Bass-Serre theory of group actions (without inversions) on a tree and integer-valued length functions on a group [2]. In the process Chiswell defined, for a group with a real-valued length function, a contractible metric space X on which there is an action of the group. For integer-valued length functions, X\s a tree in the ordinary simplicial sense. The results of our first section show that X, the metric completion of X, has the following treelike properties: (i) any two points of X are the endpoints of the image of an isometric embedding of a closed interval s u, (ii) X has no homeomorphic images of a circle, (Hi) dimZ= 1. A metric space having the properties (i), (ii), (iii) stated above will be called a tree in this article. The isometric embeddings in (i) are called geodesies. The main impetus for this investigation was to describe the structure of a group with a real-valued length function which acts freely on its associated tree; we have obtained some partial results generalizing those of Harrison [4]. Furthermore, the conjecture of Lyndon in this regard asserts that archimedean groups are isomorphic to subgroups of a free product of copies IR, with the natural length function inherited from the absolute-value function on the reals. We exhibit in §3 some counterexamples to this conjecture. Independently, J. Morgan and P. B. Shalen have proved the existence of real-valued length functions on surface groups which act freely on their associated trees. A real-valued length function on a group G is a function i.G -»• IR satisfying, for allx,j;,zeG(cf. [2]), (Al') /(l) = 0, (A2) l(x) = /(x-1), (A4) let d(x,y) = $(/(*) + l(y)- /(xr 1 )); ifd(x,y) < d{x,z) then d(y,z) = d(x,y). It is easily seen that (A4) is equivalent to saying that, if d(x, y) ^ m, d(x, z) ^ m then d(y, z) ^ m for all real numbers m. Received 25 October, 1983; revised 2 May, 1984. 1980 Mathematics Subject Classification 20E06. The research of the first author was partially supported by NSF-MCS8101585. J. London Math. Soc. (2), 31 (1985), 55-68
56
ROGER C. ALPERIN AND KENNETH N. MOSS
For a length function / there is an associated metric space X on which the group
acts. Let S = {(x,m)\xeG, meU,0^m^
l(x)} and put X= S/~ where ~ is the
equivalence relation (x,m) ~ (y,n) if and only if m = n and d{x~l,y~x) ^ m. Denote the equivalence class of (x, m) by [x, m]. Chiswell shows that A' is a metric space with metric S where M\
1r
n - / " m~n
o\\.x, m\, w,n\)-
II'
ym+n_
min{m,«}
d
^ y_^
and G acts as a group of isometries on X via x, Kg) + m-2d(x~\g)],
m>
d{x~\g).
We suggest that the reader consult Chiswell's article for a proper introduction to the theory. It is not difficult to see, for G = U * IR with its natural length function, that there are points in X— X. These can be constructed easily from summable infinite sequences of positive real numbers (cf. examples after Theorem 4.2). 1. The tree for a group with real-valued length function Chiswell has shown that if d is integer-valued then d(x, y) is the length of the geodesic from Po = [1,0] to the geodesic path from x~lP0 to .y^P,,- We show now how to construct certain canonical geodesies in X. LEMMA i. 1. Let P,QeX,d = S(P, Q); then there is an isometric embedding of the closed interval [0, d] in X, namely / : [0, d] -• X, such thatf[0) = P, f[d) = Q.
Proof. Let P = [x, m], Q = [y, n] and assume without loss of generality that m < n. We consider two cases. Case I, in which m < d{x~l,y~l). In this case d = || n—m \\ and [x,m] = \y,m] so we may define/: [0, d] -> X by f{t) = [y, m +1] for 0 < / ^ n — m. To show that this path is independent of the choice of representations for P and Q, suppose that P = [z, m] and Q = [w,n]. Notice that [z,m] = [y,m] and d(y~l, w"1) ^ n so that [y, k] = [w, k], for k ^ n; consequently [y,m + t] = [w,m + i] for 0 ^ t ^ n — m and, hence, the path is independent of the representation of its endpoints. Case II, in which m> d{x~x,y-x). and thus we define
_ I[*>m ~ & 1
[y,2d(x-\y- )-m
We have d = S(P,Q) = m +
+ t],
n-2d(x-l,y-x)
O^t^md( m-d(x~x,y~l)
This path is seen to be independent of the representation of its endpoints as follows. If P=[z,m], Q = [w,n], then d(x~x, z"1) ^ m so that [x, m — t] = [z, m — t] for 1 0 ^ f^ w - ^ " , ^ " 1 ) ; moreover, d(y~l, w'1) ^ n > d{x~x,y~l), and thus by (A4); in a similar fashion d(x~x, z~x) ^ m > d(x~x,y~x) = d(x~x,w~x), and hence d(z~x, w~x) = d(w-x,x~x) = d(x~x,y-x). Finally [y, 2d(x~x, y~x) - m +1] = [w, 2d(w~x, z"1) -m + t]
COMPLETE TREES FOR GROUPS WITH A REAL-VALUED LENGTH FUNCTION
57
for m — d(x~1,y~1) < / ^ d and thus the path is independent of the representation of its endpoints. Clearly, this is an isometric embedding on each piece 0 ^ / ^ m — d{x~l,y~l) and m-d{x-\y-1) ^ t < d. Moreover, if F = [x,m'], Q' = [y,n'],n',ml > dix^y"1) then d{P',Q') = m'+ri-2d{x-\y-1); now f[m-m') = F, f[ri+m-2d(x-\y-1)) = Q' = ri+m'-2d(x~\y-1) = d(F,Q'), so that / is and n' + m-2d{x-x,y-1)-{m-m') indeed an isometric embedding. Denote the geodesic defined above, namely/([0, d]), by PQ. Note that if S, 5" e PQ then SS' is the portion of PQ lying between S and 5". LEMMA
1.2. Let P,QeX. IfgeG then g(PQ) is a geodesic.
Proof. The argument divides into several cases and subcases. Case 1, in which n or m is less than d{x~l,y~l). We may assume that m ^ n for simplicity. The geodesic is {[y, t] \ m < / ^ «}. There are now three subcases depending on d(y~\g) = r. Subcase a, in which r ^ m < n. In this situation g(PQ) = {\gy, l(g) + t- 2d(y~\
g)]\m^t^n}
is a geodesic. Subcase b, in which r ^ n ^ m. In this situation g(PQ) = {\g, l(g) — t] \ m ^ t ^ n} is a geodesic. Subcase c, in which m ^ r ^ n. In this situation g(PQ) = {[gJ(g)-t]\m^
t^r}[)
{\gy,l(g) + t-2d(y-\g)]\r
U,
which is injective since the action is free. Furthermore, according to [6, Remark 2.2.2] the free action of a solvable group G satisfies (P3) or (P4) of [6] since a normal abelian subgroup H satisfies (P3); however condition (P4) is excluded since no abelian subgroup of G can be isomorphic to {± 1} [6, Remark 2.2.4]. PROPOSITION 2.2 (cf. [4, Corollary 5.14]). If G is an archimedean group then any two distinct maximal abelian subgroups intersect trivially.
Proof. Suppose by way of contradiction that Alt A2 are distinct maximal abelian subgroups such that 1 # xeAx n A2. From the arguments of Proposition 2.1 there are unique ends e, ex, e2 stabilized by x, Ax, A2 respectively; since xeAx, xeA2 it follows that ex = e = e2. Now choose y e Ax — A2. Since A2,y stabilize e and there is an injective homeomorphism it follows that gpC^-y) i s abelian. This contradiction means that Ax n A2 = {1}. The following are immediate corollaries, as in [4, 5.13, 5.15, 5.16]. COROLLARY 2.3. Ifx, y, z are non-trivial elements of an archimedean group G such that xy = yx and xz = zx then yz = zy. COROLLARY
2.4.
An archimedean group is a disjoint union of its maximal abelian
subgroups. COROLLARY
2.5. An archimedean group is abelian or has trivial center.
COMPLETE TREES FOR GROUPS WITH A REAL-VALUED LENGTH FUNCTION
63
3. Principal-ideal groups For a subset U of a group G with real-valued length function we define (cf. [5])
l(U)=
inf d(x,y) x, ye U
and the ideal generated by U is (U) = {z | d{z, x) ^ 1{U) for some (hence all) xe U). The subset U is called an ideal if U = (£/); it is called a principal ideal if U = (x) for some x G £/. The group G is a. principal-ideal group (PIG) if every ideal of G is principal. Let X be the associated tree to G with its length function. A point \g, m] is called a branchpoint of jfif Z - [g, m] has at least three components; the limit points of branch points will also be called branch points. 3.1. For a non-abelian group G, there is a one to one correspondence between branch points B of X and ideals U of G, which is determined by PROPOSITION
[g, m]
> U[g< m] = {z-11 [z, m] = [g, m]}.
Proof. It is immediate that U[gm] is an ideal. Suppose then that U is any ideal. Let {dix^yi)}^ be a sequence converging to /(£/). We shall show how to determine a sequence of branch points {Bt} which converge to the branch point B which is associated to U. To this end suppose that d{x,y) = /(£/); the branch point B = [x,l(U)] = \y,l(U)) separates [x, l(y)] and [y, /(£/)] and thus any element z such that l(z) ^ l(U) belongs to U. In this case B is the branch point determined by U. More generally if
...,xn,yn). then we let Bn be the branch point determined by the ideal Un = (x^y^ The sequence {B^.-^ is a sequence of branch points of decreasing distance to [1,0] and hence converges to B say, which is thus also well determined by U. The next result follows easily from the discussion above. COROLLARY 3.2. If G is a PIG then its associated tree has the property that the set of branch points coincides with the closure of{\g, l(g)] \geG}. REMARK 3.3. Since the set of branch points is closed it follows that any point in the tree of a PIG which is not a branch point lies on a unique interval with branch points for endpoints and no interior branch points.
Let 8 be the equivalence classes of the set of branch points under the natural action of G. For each beB choose a representative branch point Pb. Put Xi> = \ / Xa-> b
64
ROGER C. ALPERIN AND KENNETH N. MOSS
where Xa is a copy of X with base point Pa (here, \J denotes the coproduct in the category of sets with base points). Put Xo = X; define Xx by taking the coproduct of Xo with a copy of Xb at every branch point of Xo of type b; in general, Xn+1 is obtained from Xn by taking the coproduct of Xb at every branch point ofXn — Xn_x of type b. Let n
xO0=\jxn n-0
with the natural identification Xo c Xx c ...; it is easily seen that Xm is a tree in the sense of § 1 and also that the branch points are, in a natural sense, homogeneous. We now use this construction of X^ to obtain a PIG, namely 6, which naturally contains G and is also such that the elements of G are in one to one correspondence with the branch points of its tree Xm. For each branch point Q of A^ there is a reduced path a ( 0 from [1,0] to Q; this path is, in fact, a series of paths a o a x . . . a n where each 5 x 51 m e [0, 00)} and suppose that (1) .Fis admissible, (2) for no k > 0 and h > 0 is it the case that F(k-j) = F{k+j)~x for all 0 ^ 7 ^ h).
THEOREM
4.1. Ts is a group with an archimedean length function.
Proof. The group structure on Ts is obtained from the following multiplication scheme. Given F: [0, m] -> S x S, G: [0, n] -• S x S for F, Ce Ts let t = tFtG = sup{/i|G(k) = F(m-k)-\0
^k S x 5 for Fe Ts, then/^[Cmj-^SxSis For the proof of associativity suppose that we have F, G as before and H: [0,p] -* SxS; put tx = tF G, t2 = tG H. The argument requires three cases. Case 1, in which tx + t2rs which is length preserving. Proof. We may (and do) assume by Theorem 3.4, and the remarks following it, that G is a PIG. Let S be the set of equivalence classes of G under the equivalence relation g ~ h if and only if d(g, h) > 0. The class of g is denoted by sg; sx is the distinguished element of S. For geG, l(g) = m, define F\ [0, m] -* S x S as follows: (i) if there is heG with [h, l(h)] on the geodesic from [1,0] to \g,l(g)], then
COMPLETE TREES FOR GROUPS WITH A REAL-VALUED LENGTH FUNCTION
67
(ii) for p e [0, m] for which there is no h e G with l(h) = p there are, by Remark 3.3, unique elements hlt h2eG such that /(/ij
0 the geodesic paths from [gt, l(gt)] to [1,0], for i = 1,2, agree on the right for some positive distance. In order to show that Fg e Vs we must show that g cannot be written as g = gx g2 g^g^ where l(g) = /(g^ + Z ^ + fe^ + ^gs); this c a n be verified in a routine manner. Finally then, the homomorphism G >r g :
is a length-preserving injection; the easy details are left to the interested reader. Recall now, that a conjecture, attributed to Lyndon, asserts that any archimedean group can be embedded in a free product of copies of U preserving the natural length function l(gi • . . * „ ) = £ I I A II, i-i
where g = g1 ...gn is a word in normal form. In particular, in such groups, because of the normal form for elements in free products, the geodesic from [1,0] to [g, l(g)] in the associated tree contains only finitely many branch points {[ht, /(/»i)]}^.x with d{hi,hi+1) = 0 for i = l,...,m—1. These branch points correspond to 'changes in direction' of the geodesic. Using certain of the 'universal' archimedean groups F s we shall show that the Lyndon conjecture is false. Consider the group r s for 5" = {sv s2, sz, sit sb, s}. Let {a^ ^ 0 be an increasing m for we[0, oo) with sequence 0 = a0 < a < an < a = m. We define F:[Q,m] ^ o r some n ^ 0, 2, for some n ^ 0,
Fx(k) = = m,
a2n+1, for some n ^ 0, < a2n+2, for some n ^ 0, S,
k = m;
and define also F (n) : [0, an] -»• S x S by /r(B)l[otaB] = * V n ^ 0. In this way the points [/r(M),an], for n ^ 0, all lie on the geodesic from [1,0] to [F, m] in the tree for Fs, but d{F(n), F(n+1)) = 0, for n ^ 0. Thus we have produced the desired counterexample to the Lyndon conjecture.
References 1. L. M. BLUMENTHAL, Theory and applications of distance geometry (Clarendon Press, Oxford, 1953). 2. I. M. CHISWELL, 'Abstract length functions in groups', Math. Proc. Cambridge Philos. Soc, 80 (1976), 451-463. 3. I. M. CHISWELL, 'An example of an integer-valued length function on a group', J. London Math. Soc. (2), 16 (1977), 67-75. 3-2
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COMPLETE TREES FOR GROUPS WITH A REAL-VALUED LENGTH FUNCTION
4. N. HARRISON, 'Real length functions in groups', Trans. Amer. Math. Soc, 174 (1972), 77-106. 5. R. C. LYNDON, 'Length functions in groups', Math. Scand., 12 (1963), 209-234. 6. J. TITS, 'A theorem of Lie-Kolchin for trees', Contributions to algebra: a collection of papers dedicated to Ellis Kolchin (Adacemic Press, London, 1977), pp. 377-388.
Department of Mathematics The University of Oklahoma at Norman Oklahoma 73019, U.S.A.
Department of Defense Fort George G. Meade Maryland 20755, U.S.A.
