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Income taxes - Changes in Amount of Uncertain Tax Positions (Details) - USD ($) $ in Millions | 12 Months Ended
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Dec. 31, 2023 | Dec. 31, 2022 | Dec. 31, 2021
Income taxes |  |  |
Unrecognized tax benefits at beginning of year | $ 3.2 | $ 3.8 | $ 4.1
Net increase (decrease): |  |  |
Tax positions taken in current period | 0.5 | 0.7 | 0.6
Lapse due to applicable statute of limitations | (1.0) | (1.1) | (0.7)
Change in currency exchange rates | 0.1 | (0.2) | (0.2)
Unrecognized tax benefits at end of year | $ 2.8 | $ 3.2 | $ 3.8

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- Definition Amount of unrecognized tax benefits. + References Reference 1: http://www.xbrl.org/2003/role/disclosureRef -Topic 740 -SubTopic 10 -Name Accounting Standards Codification -Section 50 -Paragraph 15A -Subparagraph (a) -Publisher FASB -URI https://asc.fasb.org//1943274/2147482685/740-10-50-15A Reference 2: http://www.xbrl.org/2003/role/disclosureRef -Topic 740 -SubTopic 10 -Name Accounting Standards Codification -Section 45 -Paragraph 10B -Publisher FASB -URI https://asc.fasb.org//1943274/2147482525/740-10-45-10B + Details
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- Definition Amount of increase in unrecognized tax benefits resulting from tax positions that have been or will be taken in current period tax return. + References Reference 1: http://www.xbrl.org/2003/role/disclosureRef -Topic 740 -SubTopic 10 -Name Accounting Standards Codification -Section 50 -Paragraph 15A -Subparagraph (a)(2) -Publisher FASB -URI https://asc.fasb.org//1943274/2147482685/740-10-50-15A + Details
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