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Consider a group word w in n letters. For a compact group G, w induces a map G^n rightarrow G$ and thus a pushforward measure {mu}_w on G from the Haar measure on G^n. We associate to each word w a 2-dimensional cell complex X(w) and prove in Theorem 2.5 that {mu}_w is determined by the topology of X(w). The proof makes use of non-abelian cohomology and Nielsens classification of automorphisms of free groups [Nie24]. Focusing on the case when X(w) is a surface, we rediscover representation-theoretic formulas for {mu}_w that were derived by Witten in the context of quantum gauge theory [Wit91]. These formulas generalize a result of ErdH{o}s and Turan on the probability that two random elements of a finite group commute [ET68]. As another corollary, we give an elementary proof that the dimension of an irreducible complex representation of a finite group divides the order of the group; the only ingredients are Schurs lemma, basic counting, and a divisibility argument.
Alice seeks an information-theoretically secure source of private random data. Unfortunately, she lacks a personal source and must use remote sources controlled by other parties. Alice wants to simulate a coin flip of specified bias $alpha$, as a fun ction of data she receives from $p$ sources; she seeks privacy from any coalition of $r$ of them. We show: If $p/2 leq r < p$, the bias can be any rational number and nothing else; if $0 < r < p/2$, the bias can be any algebraic number and nothing else. The proof uses projective varieties, convex geometry, and the probabilistic method. Our results improve on those laid out by Yao, who asserts one direction of the $r=1$ case in his seminal paper [Yao82]. We also provide an application to secure multiparty computation.
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