{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,11]],"date-time":"2026-03-11T15:05:22Z","timestamp":1773241522155,"version":"3.50.1"},"reference-count":38,"publisher":"World Scientific Pub Co Pte Lt","issue":"06","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Algebra Comput."],"published-print":{"date-parts":[[2014,9]]},"abstract":"<jats:p> The techniques of modules and actions of groups on rooted trees are applied to study the subgroup structure and the lattice subgroup of lamplighter type groups of the form \u2112<jats:sub>n,p<\/jats:sub> = (\u2124\/p\u2124)<jats:sup>n<\/jats:sup> \u2240 \u2124 for n \u2265 1 and p prime. We completely characterize scale invariant structures on \u2112<jats:sub>1,2<\/jats:sub>. We determine all points on the boundary of binary tree (on which \u2112<jats:sub>1,p<\/jats:sub> naturally acts in a self-similar manner) with trivial stabilizer. We prove the congruence subgroup property (CSP) and as a consequence show that the profinite completion [Formula: see text] of \u2112<jats:sub>1,p<\/jats:sub> is a self-similar group generated by finite automaton. We also describe the structure of portraits of elements of \u2112<jats:sub>1,p<\/jats:sub> and [Formula: see text] and show that \u2112<jats:sub>1,p<\/jats:sub> is not a sofic tree shift group in the terminology of [T. Ceccherini-Silberstein, M. Coornaert, F. Fiorenza and Z. Sunic, Cellular automata between sofic tree shifts, Theor. Comput. Sci.506 (2013) 79\u2013101; A. Penland and Z. Sunic, Sofic tree shifts and self-similar groups, preprint]. <\/jats:p>","DOI":"10.1142\/s0218196714500374","type":"journal-article","created":{"date-parts":[[2014,9,1]],"date-time":"2014-09-01T02:13:14Z","timestamp":1409537594000},"page":"837-877","source":"Crossref","is-referenced-by-count":19,"title":["On the lattice of subgroups of the lamplighter group"],"prefix":"10.1142","volume":"24","author":[{"given":"R.","family":"Grigorchuk","sequence":"first","affiliation":[{"name":"Department of Mathematics, Texas A&amp;M University, College Station, Texas 77843, USA"}]},{"given":"R.","family":"Kravchenko","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Chicago, Chicago, Illinois 60637, USA"}]}],"member":"219","published-online":{"date-parts":[[2014,11,16]]},"reference":[{"key":"rf1","doi-asserted-by":"publisher","DOI":"10.1112\/S002460930500425X"},{"key":"rf2","doi-asserted-by":"publisher","DOI":"10.4171\/GGD\/124"},{"key":"rf4","unstructured":"M. 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