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Amer. Math. Soc."],"abstract":"<p>\n                    The set of finitely generated subgroups of the group\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper P upper L Subscript plus Baseline left-parenthesis upper I right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>P<\/mml:mi>\n                            <mml:msub>\n                              <mml:mi>L<\/mml:mi>\n                              <mml:mo>+<\/mml:mo>\n                            <\/mml:msub>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>I<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">PL_+(I)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    of orientation-preserving piecewise-linear homeomorphisms of the unit interval includes many important groups, most notably R.\u00a0Thompson\u2019s group\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper F\">\n                        <mml:semantics>\n                          <mml:mi>F<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">F<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . Here, we show that every finitely generated subgroup\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper G greater-than upper P upper L Subscript plus Baseline left-parenthesis upper I right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>G<\/mml:mi>\n                            <mml:mo>&gt;<\/mml:mo>\n                            <mml:mi>P<\/mml:mi>\n                            <mml:msub>\n                              <mml:mi>L<\/mml:mi>\n                              <mml:mo>+<\/mml:mo>\n                            <\/mml:msub>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>I<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">G&gt;PL_+(I)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    is either soluble, or contains an embedded copy of the finitely generated, non-soluble Brin-Navas group\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper B\">\n                        <mml:semantics>\n                          <mml:mi>B<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">B<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , affirming a conjecture of the first author from 2009. In the case that\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper G\">\n                        <mml:semantics>\n                          <mml:mi>G<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">G<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    is soluble, we show the derived length of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper G\">\n                        <mml:semantics>\n                          <mml:mi>G<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">G<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    is bounded above by the number of breakpoints of any finite set of generators. We specify a set of \u2018computable\u2019 subgroups of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper P upper L Subscript plus Baseline left-parenthesis upper I right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>P<\/mml:mi>\n                            <mml:msub>\n                              <mml:mi>L<\/mml:mi>\n                              <mml:mo>+<\/mml:mo>\n                            <\/mml:msub>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>I<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">PL_+(I)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    (which includes R. Thompson\u2019s group\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper F\">\n                        <mml:semantics>\n                          <mml:mi>F<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">F<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    ) and give an algorithm which determines whether or not a given finite subset\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper X\">\n                        <mml:semantics>\n                          <mml:mi>X<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">X<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    of such a computable group generates a soluble group. When the group is soluble, the algorithm also determines the derived length of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"mathematical left-angle upper X mathematical right-angle\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mo fence=\"false\" stretchy=\"false\">\n                              \u27e8\n                              \n                            <\/mml:mo>\n                            <mml:mi>X<\/mml:mi>\n                            <mml:mo fence=\"false\" stretchy=\"false\">\n                              \u27e9\n                              \n                            <\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\langle X\\rangle<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . Finally, we give a solution of the membership problem for a particular family of finitely generated soluble subgroups of any computable subgroup of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper P upper L Subscript plus Baseline left-parenthesis upper I right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>P<\/mml:mi>\n                            <mml:msub>\n                              <mml:mi>L<\/mml:mi>\n                              <mml:mo>+<\/mml:mo>\n                            <\/mml:msub>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>I<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">PL_+(I)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    .\n                  <\/p>","DOI":"10.1090\/tran\/8421","type":"journal-article","created":{"date-parts":[[2021,3,31]],"date-time":"2021-03-31T12:22:49Z","timestamp":1617193369000},"source":"Crossref","is-referenced-by-count":1,"title":["Determining solubility for finitely generated groups of PL homeomorphisms"],"prefix":"10.1090","author":[{"given":"Collin","family":"Bleak","sequence":"first","affiliation":[]},{"given":"Tara","family":"Brough","sequence":"additional","affiliation":[]},{"given":"Susan","family":"Hermiller","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2021,7,19]]},"reference":[{"key":"1","unstructured":"Thompson\u2019s group at 40 years, Preliminary Problem List, 2004, available at \\url{http:\/\/www.aimath.org\/WWN\/thompsonsgroup\/thompsonsgroup.pdf}."},{"key":"2","doi-asserted-by":"publisher","first-page":"239","DOI":"10.1007\/s10711-013-9853-2","article-title":"Conjugacy and dynamics in Thompson\u2019s groups","volume":"169","author":"Belk, James","year":"2014","journal-title":"Geom. 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(2)","ISSN":"https:\/\/id.crossref.org\/issn\/0024-6107","issn-type":"print"},{"issue":"1","key":"5","doi-asserted-by":"publisher","first-page":"1","DOI":"10.4171\/GGD\/50","article-title":"A minimal non-solvable group of homeomorphisms","volume":"3","author":"Bleak, Collin","year":"2009","journal-title":"Groups Geom. Dyn.","ISSN":"https:\/\/id.crossref.org\/issn\/1661-7207","issn-type":"print"},{"issue":"2","key":"6","doi-asserted-by":"publisher","first-page":"449","DOI":"10.1112\/S0024610799007905","article-title":"The ubiquity of Thompson\u2019s group \ud835\udc39 in groups of piecewise linear homeomorphisms of the unit interval","volume":"60","author":"Brin, Matthew G.","year":"1999","journal-title":"J. London Math. Soc. (2)","ISSN":"https:\/\/id.crossref.org\/issn\/0024-6107","issn-type":"print"},{"issue":"4","key":"7","doi-asserted-by":"publisher","first-page":"619","DOI":"10.1142\/S0218196705002517","article-title":"Elementary amenable subgroups of R. 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