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Let X be a universal cover of a finite connected graph, G = Aut(X), and F a group acting discretely and cocompactly on X, i.e., a uniform lattice on X. We give a necessary condition for an elliptic element of G to belong to a uniform lattice or to the commensurability group. By using this condition, we construct some explicit examples. Continuing the classic Bass-Serre theory on graphs of groups [S], Bass developed the covering theory for graphs of groups [B]. Using this, Bass and Kulkarni developed the uniform tree lattices theory in their joint paper [BK]. In that paper they obtained a lot of important results. It is fruitful to think of (G, X, F) as a combinatorical analogue of (PSL2(R), upper half plane, fuchsian group). Let X be a 'uniform tree', i.e., the universal cover of a finite connected graph, G = Aut(X), equipped with compact open topology. The subgroup H < G is discrete iff every vertex stabilizer H, for x E VX is finite, where VX is the, set of all vertices of X. We call F < G a uniform X-lattice if F is discrete and the quotient graph F\X is finite (i.e., VX has only finitely many F-orbits). Let Fo, Fi be subgroups of G. F0 and F1 are said to be commensurable (denoted FO F ), if the index [ri: FO n rF] is finite for i = 0, 1 . The commensurator (or "virtual normalizer') of F in G is the group CG(F) = {g E GIgFg-I r} . It was shown in [BK] that, up to G-conjugacy, any two uniform lattices in G are commensurable. Thus the commensurator CG(F) of a uniform lattice F < G is, up to conjugacy, independent of F; we denote it by C(X). It is proved in [LI] that C(X) is dense in G, which was conjectured in [BK]. In this paper, we give a necessary condition for an elliptic element (i.e., one having fixed points) of G to belong to a uniform lattice or to the commensurability group C(X). By this condition, it is then easy to construct some automorphisms of X which do not belong to a uniform lattice, nor do they belong to C(X). We address here the following questions: Question. Let X be a uniform tree, G = Aut(X), g E G. When is there a uniform X-lattice F: (a) such that g E IF?; (b) such that g E CG(F) ? Received by the editors July 19, 1992. 1991 Mathematics Subject Classification. Primary 05C25, 05C05. ? 1994 American Mathematical Society 0002-9939/94 $1.00 + $.25 per page

A necessary condition for an elliptic element to belong to a uniform tree lattice