Question: Say a group of people vote for a leader in an election but the leader then goes on to "*speak for the people*" but actually he's speaking for himself and his cronies. I'd like it to be clearly in a derogatory sense.

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> "Mr ..... claimed he was speaking on behalf of the people who voted for him but he was actually **[stealing]** their voice"
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What word would best fit here?

Answer: 
You could use the term *co-opt*:

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> **co-opt** *v. tr.*  
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>  3. To take or assume for one's own use; appropriate: *co-opted the criticism by embracing it.*
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[From TFDO](http://www.thefreedictionary.com/co-opt)
How about *pervert*?

From Google search:

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> [pervert](https://www.google.com/search?client=safari&rls=en&q=pervert+definition&ie=UTF-8&oe=UTF-8) *verb*:
>  alter (something) from its original course, meaning, or state to a
>  distortion or corruption of what was first intended
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Your example:

> 
> Mr ..... claimed he was speaking on behalf of the people who voted for
>  him but he was actually *perverting* their voice
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> 

It's hard to find a verb more derogatory than *pervert*, not to mention its other obvious connotations as a noun.

Also, as suggested by @alwayslearning, how about *misappropriate*?

From Dictionary.com:

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> [*misappropriate*](http://www.dictionary.com/browse/misappropriate?s=t):
>  to put to a wrong use
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Your example:

> 
> Mr ..... claimed he was speaking on behalf of the people who voted for
>  him but he was actually *misappropriating* their voice.
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>