Finitely generated subgroups of free groups as formal languages and
their cogrowth
Keyword(s):
For finitely generated subgroups $H$ of a free group $F_m$ of finite rank $m$, we study the language $L_H$ of reduced words that represent $H$ which is a regular language. Using the (extended) core of Schreier graph of $H$, we construct the minimal deterministic finite automaton that recognizes $L_H$. Then we characterize the f.g. subgroups $H$ for which $L_H$ is irreducible and for such groups explicitly construct ergodic automaton that recognizes $L_H$. This construction gives us an efficient way to compute the cogrowth series $L_H(z)$ of $H$ and entropy of $L_H$. Several examples illustrate the method and a comparison is made with the method of calculation of $L_H(z)$ based on the use of Nielsen system of generators of $H$.
2016 ◽
Vol 26
(05)
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pp. 843-886
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2007 ◽
Vol 17
(08)
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pp. 1493-1535
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2006 ◽
Vol 16
(06)
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pp. 1031-1045
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2019 ◽
Vol 30
(06n07)
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pp. 1197-1216
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1971 ◽
Vol 5
(1)
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pp. 87-94
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2012 ◽
Vol 22
(04)
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pp. 1250030
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