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What dialect is this: omitting a hard "t" in a word such as button. Sounds like they're saying buh en.

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This sounds like glottal replacement. It happens in several dialects. According to Wikipedia:

In RP, and in many accents such as Cockney as well as all American English, it is common for /t/ to be completely replaced by a glottal stop before another consonant, as in not now [nɒʔnaʊ] and department [dɪpʰɑː(ɹ)ʔmən̩t]. In General American (GA) English, this replacement also happens before a syllabic /n/, as in button (representable as [ˈbʌʔn̩] or [ˈbʌʔɪ̈n]).

This page has good examples in a British accent.

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