My solid state relay seems to be dropping the potential by 35 volts. I read about 80 volts from one end of a modified plug and the other end is plugged into a 120 volt wall socket. I have the hot wire cut and the ends connected to the leads of the relay. The neutral wire is not cut anywhere. I knew there would be a small internal resistance but not by this much. Is there something I am missing? I would simply like to control a 120 volt lamp with my arduino.
More than likely what you are seeing is the leakiness of the SSR when it is switched off. Lets say you meter has 10 Mohm input impedance - the 80V implies a current of 8 uA which in turn implies the SSR has an open-circuit impedance of 35 v / 8 uA = 4.38 Mohms.
Else, the SSR is broken!
Zero-voltage switching type thyristor SSRs sometimes require a minimum load current.
The gate trigger pulse for the thyristor is generated near the mains zero crossing. If there is insufficient load to reach the minimum holding current (of the pilot device in the case of an opto-triac), then the pilot thyristor switches back off until the next zero crossing- the gate trigger does not persist past the window around the mains zero crossing. Since the holding current of the main triac will be much higher than the pilot triac, the SSR will only turn on for a brief time when the mains voltage is very low. If you've got a meter connected, you'll measure leakage current through the triacs (and possibly a snubber circuit).
Take the MOC0603 opto-triac output optocoupler. This device must see a current exceeding its holding current of 0.5mA before the mains voltage exceeds about +/-12V or the opto-triac will turn off again and never trigger the main triac. Below is a simplified schematic of an SSR with a zero-crossing circuit.
That means that the load resistance RL must be less than 24K ohms (typically, at room temperature) for the SSR to turn on for the full cycle.
If you connect a reasonable load to the SSR (there may be a specification in the datasheet, 100mA is a common requirement for minimum load) then you should measure mains voltage across the load when the input is "on", less a volt or two.