So GN z-11 is the latest "furthest away" galaxy. It has a claimed redshift measurement of $z=11.1$, meaning that we are seeing the light it emitted about 400 Myr after the big bang (dependent on an assumed set of cosmological parameters).
To get the redshift measurement, the discoverers used grism (relatively low resolution) spectroscopy in the near infrared. What they were looking for is the rest-frame Lyman alpha continuum break, which would be redshifted into this wavelength range.
The Lyman alpha continuum break is caused by the absorption of nearly all ultraviolet photons with wavelengths smaller than 121 nm. These high energy photons are capable of photoionising the neutral hydrogen present in the intergalactic medium at a range of lower redshifts. This neutral hydrogen is present in abundance at redshifts greater than 6, as it had not yet been re-ionised by quasars and starlight. The consequence is that no light is expected to reach us from rest-frame wavelengths shortwards of 121 nm, but a galaxy's light can reach us from rest-frame wavelengths that are longer than this. When one observes the spectrum, we see flux at long wavelengths which suddenly cuts off at shorter wavelengths. The wavelength of the break is $lambda = 121 times (1+z)$ nm, where $z$ is the redshift.
It is this Lyman alpha continuum break that has been identified at an observed wavelength of 1470 nm. This leads to the redshift estimate of $z= (1470/121)-1 = 11.1$.
The details are presented in Oesch et al. (2016). The continuum break is the only thing visible in the spectrum. The authors are confident that this is what it is because their previous broadband photometry had given them an estimated redshift of $>10$ (which is why they observed this candidate in the first place).
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