To be precise, the point of PoW is not having a recurring cost, but just organizing a "decentralized lottery" where it is very unlikely that you get to verify your own (fake) transactions.

As someone commented above, there are studies on "proof of idle", but other systems have bigger drawbacks than PoW.

I'm not sure what you mean by fake transactions, but transactions are verified with public key cryptography, and a block with invalid transactions will be rejected out of hand by blockchain nodes.

There are two* major classes of attack which someone could do with the power to mint blocks at will (i.e. a 51% attack):

    - discriminatory filtering, where valid transactions are 
      never validated
    - double spends, where a transaction appears like it's 
      been validated for a time but is then rolled back and 
      becomes invalid

But no one can mint transactions without private keys.

* They can also monopolize the rewards of mining, but that's not nearly as bad as the other issues.