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Specifically, we show that when [Formula: see text], the groups [Formula: see text] have property [Formula: see text], that is, every automorphism has an infinite number of twisted conjugacy classes. In contrast, when [Formula: see text] the lamplighter groups [Formula: see text] have property [Formula: see text] if and only if [Formula: see text]. <\/jats:p>","DOI":"10.1142\/s0218196715500411","type":"journal-article","created":{"date-parts":[[2015,11,26]],"date-time":"2015-11-26T08:33:49Z","timestamp":1448526829000},"page":"1275-1299","source":"Crossref","is-referenced-by-count":9,"title":["Automorphisms of higher rank lamplighter groups"],"prefix":"10.1142","volume":"25","author":[{"given":"Melanie","family":"Stein","sequence":"first","affiliation":[{"name":"Department of Mathematics, Trinity College, Hartford, CT 06106, USA"}]},{"given":"Jennifer","family":"Taback","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Bowdoin College, Brunswick, ME 04011, USA"}]},{"given":"Peter","family":"Wong","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Bates College, Lewiston, ME 04240, USA"}]}],"member":"219","published-online":{"date-parts":[[2015,12,30]]},"reference":[{"key":"S0218196715500411BIB002","doi-asserted-by":"publisher","DOI":"10.4171\/JEMS\/130"},{"key":"S0218196715500411BIB003","doi-asserted-by":"publisher","DOI":"10.1007\/s00041-005-3079-0"},{"key":"S0218196715500411BIB004","doi-asserted-by":"publisher","DOI":"10.1017\/S1446788700007783"},{"key":"S0218196715500411BIB005","series-title":"Graduate Texts in Mathematics","volume-title":"Cohomology of Groups","volume":"87","author":"Brown K. 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