LOCAL LIMIT THEOREMS FOR FREE GROUPS Richard Sharp University of Manchester Abstract. In this paper we obtain a local limit theorem for elements of a free group G under the abelianization map [·]: G → G/[G, G]. This is obtained via an analysis involving subshifts of finite type, where we obtain a result of independent interest. The case of fundamental groups of compact surfaces of genus ≥ 2 is also discussed. 0. Introduction Let G denote the free group on k ≥ 2 generators {a 1 ,... ,a k }. For g ∈ G, let |g| denote its word length, i.e., |g| = inf {n ≥ 0: g = g 1 ··· g n ,g i ∈{a ±1 1 ,... ,a ±1 k }}, and let [g] denote the image of g under the abelianization map [·]: G → G/[G, G] ∼ = Z k . Let W(n)= {g ∈ G : |g| = n} and observe that #W(n)=2k(2k − 1) n−1 . In this paper, we shall be interested in the distribution of the elements of W(n) in Z k via the mapping [·], as n →∞. In particular, defining W(n, α)= {g ∈W(n):[g]= α}, we wish to examine the dependence of #W(n, α) on α as well as on n. Our approach is to regard #W(n, α)/#W(n) as a probability distribution on Z k and to ask about its limiting behaviour as n →∞. Rivin has shown that a central limit theorem is satisfied, i.e., for A ⊂ R k , lim n→∞ 1 #W(n) #{g ∈W(n):[g]/ √ n ∈ A} = 1 (2π) k/2 σ k  A e −||x|| 2 /2σ 2 dx, where || · || denotes the Euclidean norm and where σ 2 = 1 √ 2k − 1  1+  k + √ 2k − 1 k − √ 2k − 1  1/2  (0.1) [18]. (In fact, this result is similar in spirit to earlier results for subshifts of finite type, hyperbolic diffeomorphisms, and interval maps [1], [4], [5], [10], [12], [17], [19], [20], [23].) Here, we shall establish a more precise local limit theorem. First we note a combinatorial restriction. We shall say that α =(α 1 ,... ,α k ) is even if α 1 + ··· + α k is even, and odd otherwise. It is clear that if [g]= α then α has the same parity as |g|. Thus, in particular, The author was supported by an EPSRC Advanced Research Fellowship. Typeset by A M S-T E X 1