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- W1994395900 abstract "We show how a positive solution to the word problem for finitely generated commutative semigroups leads to a positive solution to the conjugacy problem for a class of HNN groups. Here we answer a question of Professor Wilhelm Magnus concerning the solvability of the conjugacy problem of the groups VA discussed in [1]. These groups are given by (I) Ka-l, . . where piqi > 1 and (pi, qi) = 1. We call the groups given by (I) VA -groups and denote them by G(p1, q 1, . . . ,Pk, qk) as in [1]. We call the integers appearing in (I) the exponents of the group. Let / and m be nonzero integers and call m reachable from / with respect to the exponents if there is a sequence of integers beginning with I and ending with m, such that for successive terms 1I and l+,j either lj+j = li(qj/pj) or lj+j = li(pj/qj). The reachability problem for the exponents is to decide for arbitrary such I and m whether m is reachable from 1. From [1, Lemma 1] we obtain LEMMA 1. A VA -group has solvable conjugacy problem if and only if the reachability problem for its exponents is solvable. The reachability problem for the exponents of VA-group is equivalent to the reachability problem for a class of self-dual vector addition systems [1]. Recently several computer scientists have solved the reachability problem for self-dual vector addition systems using combinatorial methods. Below we give a simple self-contained algebraic proof of the solvability of the reachability problem for the exponents. This allows us to prove THEOREM. VA -groups have solvable conjugacy problem. It suffices to consider the reachability problem restricted to the set M of positive integers whose prime divisors are among the prime divisors of the exponents p1, q1, ... Pk, qk. Observe that M forms a multiplicative semigroup and the reachability relation induces a congruence on M. Denote the Received by the editors April 23, 1976. AMS (MOS) subject classifications (1970). Primary 20E05, 20F05; Secondary 20M05. O American Mathematical Society 1977" @default.
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- W1994395900 date "1976-02-01" @default.
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- W1994395900 title "The conjugacy problem for HNN groups and the word problem for commutative semigroups" @default.
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- W1994395900 doi "https://doi.org/10.1090/s0002-9939-1976-0422457-9" @default.
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