A broken flexible pump coupler, clogged hydraulic fluid filter, failed hydraulic control valve, broken hydraulic pump or faulty hydraulic cylinder can prevent the hydraulic cylinder from extending the wedge on a log splitter.
Accessing the flexible pump coupler is fairly easy, so check that part first. The flexible pump coupler connects the engine drive shaft to the hydraulic pump. The coupler is designed to break if the hydraulic pump seizes up, protecting the engine from damage. Because the flexible coupler is plastic, the coupler can deteriorate over time and crack or break. Replace the flexible coupler if it's broken or damaged.
If the flexible coupler broke, try to rotate the hydraulic pump drive shaft with the drive shaft disconnected from the engine drive shaft to check for a locked-up hydraulic pump. If you can't rotate the hydraulic pump shaft, you'll likely need to replace the hydraulic pump. You'll also need to replace the hydraulic pump if it's leaking hydraulic fluid because the pump can't build up hydraulic fluid pressure to extend the wedge to split a piece of wood if its seals leak.
Hydraulic fluid leaking from the hydraulic control valve or hydraulic cylinder often indicates that the leaking part has failed. You'll likely need to replace the control valve and/or cylinder if they leak hydraulic fluid.
Although it doesn't happen often, a clogged hydraulic fluid filter can impede hydraulic fluid flow and prevent the cylinder from extending the wedge. Check the hydraulic fluid filter and replace the filter if it's clogged.