Two Examples Related to the Twisted Burnside–Frobenius Theory for Infinitely Generated Groups
TheTBFT f conjecture, which is a modification of a conjecture by Fel’shtyn and Hill, says that if the Reidemeister number R ( φ ) of an automorphism φ of a (countable discrete) group G is finite, then it coincides with the number of fixed points of the corresponding homeomorphism φ ̂ of G ̂ f (the p...
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| Veröffentlicht in: | Journal of mathematical sciences (New York, N.Y.) N.Y.), 2020-08, Vol.248 (5), p.661-666 |
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| container_title | Journal of mathematical sciences (New York, N.Y.) |
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| creator | Troitsky, E. V. |
| description | TheTBFT
f
conjecture, which is a modification of a conjecture by Fel’shtyn and Hill, says that if the Reidemeister number
R
(
φ
) of an automorphism
φ
of a (countable discrete) group
G
is finite, then it coincides with the number of fixed points of the corresponding homeomorphism
φ
̂
of
G
̂
f
(the part of the unitary dual formed by finite-dimensional representations). The study of this problem for residually finite groups has been the subject of some recent activity. We prove here that for infinitely generated residually finite groups there are positive and negative examples for this conjecture. It is detected that the finiteness properties of the number of fixed points of
φ
itself also differ from the finitely generated case. |
| doi_str_mv | 10.1007/s10958-020-04903-0 |
| format | Article |
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f
conjecture, which is a modification of a conjecture by Fel’shtyn and Hill, says that if the Reidemeister number
R
(
φ
) of an automorphism
φ
of a (countable discrete) group
G
is finite, then it coincides with the number of fixed points of the corresponding homeomorphism
φ
̂
of
G
̂
f
(the part of the unitary dual formed by finite-dimensional representations). The study of this problem for residually finite groups has been the subject of some recent activity. We prove here that for infinitely generated residually finite groups there are positive and negative examples for this conjecture. It is detected that the finiteness properties of the number of fixed points of
φ
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f
conjecture, which is a modification of a conjecture by Fel’shtyn and Hill, says that if the Reidemeister number
R
(
φ
) of an automorphism
φ
of a (countable discrete) group
G
is finite, then it coincides with the number of fixed points of the corresponding homeomorphism
φ
̂
of
G
̂
f
(the part of the unitary dual formed by finite-dimensional representations). The study of this problem for residually finite groups has been the subject of some recent activity. We prove here that for infinitely generated residually finite groups there are positive and negative examples for this conjecture. It is detected that the finiteness properties of the number of fixed points of
φ
itself also differ from the finitely generated case.</description><subject>Automorphisms</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><issn>1072-3374</issn><issn>1573-8795</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2020</creationdate><recordtype>article</recordtype><recordid>eNqNks1qGzEUhYfSQlO3L9DVQFddKL2SRiNpmYbENQQKqbsW49GVozAeudIMiXd5h75hn6SKXYgNJg1aSFd85xz93KL4SOGUAsgviYIWigADApUGTuBVcUKF5ERJLV7nNUhGOJfV2-JdSreQRbXiJ4WZ34Xy4r5ZrTtM5TV2zYC2HEI53GA5v_Ppsfw6xj55i38efl_GsMDej6mc32CIm9KFWM5653s_YLcpp9hj3HpMYxjX6X3xxjVdwg__5knx8_Jifv6NXH2fzs7Prkhb1RzIAp2oXFPX1GpqQahatqritGKCWYmaK8pQLqCV1FnnrMLWWiWso5XSjGs-KT7tfNcx_BoxDeY25FPnSMMqBpJWWtAnatl0aHzvwhCbduVTa85qxZgAqOvnKa5ZfuWcOSnIEWq5vX8XenQ-bx-4vojf8z89wudhceXbowEvE-wlfD4QZGbA-2HZjCmZ2Y_rQ_P_snu-bMe2MaQU0Zl19KsmbgwF89iuZteuJuNm264GsojvRCnD_RLj0wc-o_oL-_voCw</recordid><startdate>20200801</startdate><enddate>20200801</enddate><creator>Troitsky, E. V.</creator><general>Springer US</general><general>Springer</general><general>Springer Nature B.V</general><scope>AAYXX</scope><scope>CITATION</scope><scope>ISR</scope></search><sort><creationdate>20200801</creationdate><title>Two Examples Related to the Twisted Burnside–Frobenius Theory for Infinitely Generated Groups</title><author>Troitsky, E. V.</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c4630-bef54fa661d91d05867c84314252d7e93812e7b0c71fdffd8ecdd85df14892393</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2020</creationdate><topic>Automorphisms</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Troitsky, E. V.</creatorcontrib><collection>CrossRef</collection><collection>Gale In Context: Science</collection><jtitle>Journal of mathematical sciences (New York, N.Y.)</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Troitsky, E. V.</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Two Examples Related to the Twisted Burnside–Frobenius Theory for Infinitely Generated Groups</atitle><jtitle>Journal of mathematical sciences (New York, N.Y.)</jtitle><stitle>J Math Sci</stitle><date>2020-08-01</date><risdate>2020</risdate><volume>248</volume><issue>5</issue><spage>661</spage><epage>666</epage><pages>661-666</pages><issn>1072-3374</issn><eissn>1573-8795</eissn><abstract>TheTBFT
f
conjecture, which is a modification of a conjecture by Fel’shtyn and Hill, says that if the Reidemeister number
R
(
φ
) of an automorphism
φ
of a (countable discrete) group
G
is finite, then it coincides with the number of fixed points of the corresponding homeomorphism
φ
̂
of
G
̂
f
(the part of the unitary dual formed by finite-dimensional representations). The study of this problem for residually finite groups has been the subject of some recent activity. We prove here that for infinitely generated residually finite groups there are positive and negative examples for this conjecture. It is detected that the finiteness properties of the number of fixed points of
φ
itself also differ from the finitely generated case.</abstract><cop>New York</cop><pub>Springer US</pub><doi>10.1007/s10958-020-04903-0</doi><tpages>6</tpages></addata></record> |
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| ispartof | Journal of mathematical sciences (New York, N.Y.), 2020-08, Vol.248 (5), p.661-666 |
| issn | 1072-3374 1573-8795 |
| language | eng |
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| source | SpringerLink Journals - AutoHoldings |
| subjects | Automorphisms Mathematics Mathematics and Statistics |
| title | Two Examples Related to the Twisted Burnside–Frobenius Theory for Infinitely Generated Groups |
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