{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,4]],"date-time":"2025-07-04T00:40:06Z","timestamp":1751589606488,"version":"3.41.0"},"reference-count":0,"publisher":"Centre pour la Communication Scientifique Directe (CCSD)","license":[{"start":{"date-parts":[[2025,7,4]],"date-time":"2025-07-04T00:00:00Z","timestamp":1751587200000},"content-version":"am","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2025,7,4]],"date-time":"2025-07-04T00:00:00Z","timestamp":1751587200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2025,7,4]],"date-time":"2025-07-04T00:00:00Z","timestamp":1751587200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"accepted":{"date-parts":[[2025,7,2]]},"abstract":"<jats:p>For any group $G$ and integer $k\\ge 2$ the Andrews-Curtis transformations act as a permutation group, termed the Andrews-Curtis group $AC_k(G)$, on the subset $N_k(G) \\subset G^k$ of all $k$-tuples that generate $G$ as a normal subgroup (provided $N_k(G)$ is non-empty). The famous Andrews-Curtis Conjecture is that if $G$ is free of rank $k$, then $AC_k(G)$ acts transitively on $N_k(G)$. The set $N_k(G)$ may have a rather complex structure, so it is easier to study the full Andrews-Curtis group $FAC(G)$ generated by AC-transformations on a much simpler set $G^k$. Our goal here is to investigate the natural epimorphism $\u03bb\\colon FAC_k(G) \\to AC_k(G)$. We show that if $G$ is non-elementary torsion-free hyperbolic, then $FAC_k(G)$ acts faithfully on every nontrivial orbit of $G^k$, hence $\u03bb\\colon FAC_k(G) \\to AC_k(G)$ is an isomorphism.<\/jats:p><jats:p>7 pages. In memory of Ben Fine. Published in journal of Groups, Complexity, Cryptology<\/jats:p>","DOI":"10.46298\/jgcc.2025..15972","type":"journal-article","created":{"date-parts":[[2025,7,4]],"date-time":"2025-07-04T00:15:12Z","timestamp":1751588112000},"source":"Crossref","is-referenced-by-count":0,"title":["Andrews-Curtis groups"],"prefix":"10.46298","volume":"Volume 16, Issue 1, Special...","author":[{"given":"Robert H.","family":"Gilman","sequence":"first","affiliation":[{"id":[{"id":"https:\/\/ror.org\/02z43xh36","id-type":"ROR","asserted-by":"publisher"}],"name":"Stevens Institute of Technology"}]},{"given":"Alexei G.","family":"Myasnikov","sequence":"additional","affiliation":[{"id":[{"id":"https:\/\/ror.org\/02z43xh36","id-type":"ROR","asserted-by":"publisher"}],"name":"Stevens Institute of Technology"}]}],"member":"25203","published-online":{"date-parts":[[2025,7,4]]},"container-title":["journal of Groups, complexity, cryptology"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/arxiv.org\/pdf\/2506.23031v2","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/arxiv.org\/pdf\/2506.23031v2","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,7,4]],"date-time":"2025-07-04T00:15:12Z","timestamp":1751588112000},"score":1,"resource":{"primary":{"URL":"https:\/\/gcc.episciences.org\/15972"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,7,4]]},"references-count":0,"URL":"https:\/\/doi.org\/10.46298\/jgcc.2025..15972","relation":{"is-same-as":[{"id-type":"arxiv","id":"2506.23031","asserted-by":"subject"},{"id-type":"doi","id":"10.48550\/arXiv.2506.23031","asserted-by":"subject"}]},"ISSN":["1869-6104"],"issn-type":[{"value":"1869-6104","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,7,4]]}}}