ON A FORMULA FOR THE ASYMPTOTIC DIMENSION OF FREE PRODUCTS G. C. BELL, A. N. DRANISHNIKOV, AND J. E. KEESLING Abstract. We prove an exact formula for the asymptotic dimension of a free product. Our main theorem states that if A and B are finitely gen- erated groups and with asdim A = n, and asdim B ≤ n then asdim(A * B) = max{n, 1}. 1. Introduction The asymptotic dimension of a metric space was defined by Gromov in [9] in his study of asymptotic properties of finitely generated groups. The asymptotic dimension of the finitely generated group Γ is defined to be the asymptotic dimension of the metric space |Γ| S associated to a finite gener- ating set S. The metric is the word metric, i.e., the maximal left-invariant metric with respect to the property that dist(s,e) = 1 and dist(s −1 ,e)=1, for all s ∈ S. Two finite generating sets give rise to Lipschitz equivalent met- rics. As asdim is a coarse invariant (so in particular an invariant of Lipschitz equivalence), asdim Γ is well-defined without reference to a generating set. In [13], Yu proved that if Γ has finite asymptotic dimension then the coarse Baum-Connes conjecture and hence the Novikov higher signature conjecture hold for Γ. This result meant that a lot of work was devoted to showing the finiteness of asdim for various groups, (see, for instance, [7], [4], and [10]). The first two authors showed in [1] that the finiteness of asdim was pre- served by the construction of the amalgamated free product of groups. The result was concerned with proving finite asdim and so the estimate given there is quite rough. This estimate was improved in [2], and examples were given to show that the upper bound could not be improved. The purpose of this note is to prove a formula for the asdim of a free product. This proof of this result uses the asymptotic analog of large in- ductive dimension, called asymptotic inductive dimension, asInd, defined by the second author, in [5]. In particular we prove that asdim A ∗ B = max{asdim A, asdim B, 1}. This formula agrees with the conjectured formula for the free product with amalgamation given in [2]. In §2 we recall the definitions of asdim and asInd. In the next section, we recall some technical results which are needed for the proof of the formula. The formula itself appears as the main theorem in the final section. 2000 Mathematics Subject Classification. Primary: 20F69, Secondary: 20E08, 20E06. 1