TY - JOUR
T1 - Multiplicative measures on free groups
AU - Borovik, Alexandre V.
AU - Myasnikov, Alexei G.
AU - Remeslennikov, Vladimir N.
PY - 2003/12
Y1 - 2003/12
N2 - We introduce a family of multiplicative distributions {μs|s ∈ (0, 1)} on a free group F and study it as a whole. In this approach, the measure of a given set R ⊆ F is a function μ(R): s → μ s(R), rather then just a number. This allows one to evaluate sizes of sets using analytical properties of their measure functions μ(R). We suggest a new hierarchy of subsets R in F with respect to their size, which is based on linear approximations of the function μ(R). This hierarchy is quite sensitive, for example, it allows one to differentiate between sets with the same asymptotic density. Estimates of sizes of various subsets of F are given.
AB - We introduce a family of multiplicative distributions {μs|s ∈ (0, 1)} on a free group F and study it as a whole. In this approach, the measure of a given set R ⊆ F is a function μ(R): s → μ s(R), rather then just a number. This allows one to evaluate sizes of sets using analytical properties of their measure functions μ(R). We suggest a new hierarchy of subsets R in F with respect to their size, which is based on linear approximations of the function μ(R). This hierarchy is quite sensitive, for example, it allows one to differentiate between sets with the same asymptotic density. Estimates of sizes of various subsets of F are given.
KW - Asymptotic density
KW - Free groups
KW - Generating functions
KW - Growth functions
KW - Measures
KW - Negligible sets
KW - Random walks
KW - Recurrent and amenable groups
KW - Regular sets
KW - Sparse
KW - Thick
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U2 - 10.1142/S0218196703001596
DO - 10.1142/S0218196703001596
M3 - Article
AN - SCOPUS:0442291619
SN - 0218-1967
VL - 13
SP - 705
EP - 731
JO - International Journal of Algebra and Computation
JF - International Journal of Algebra and Computation
IS - 6
ER -