The work in this thesis uses 38 orbits worth of Hubble Space Telescope (HST), Wide Field Camera 3 (WFC3) infrared F140W imaging and G141 grism data on 10 galaxy clusters at z∼1 to study how the quenching of star formation in galaxies and galaxy size growth operates in different environments at this redshift. The unique capabilities of space-based slitless spectroscopy with the G141 grism allows for the construction of spatially resolved Hα maps, providing the possibility to directly observe environmental quenching at z∼1 for the first time. This allows us to understand the detailed physics behind environmental quenching mechanisms and how they operate on galaxies at this high redshift.

The quenching of star formation leads to the build-up of quenched (or ``quiescent”) galaxies in the Universe. Observations have also shown that quiescent galaxies grow disproportionately more in size than stellar mass from high to low redshifts. Many studies have argued that minor mergers are responsible for this size growth. To test this hypothesis, it is possible to use the cluster environment as a laboratory. Cluster galaxies have high peculiar velocities, making mergers between them rare. Since minor mergers are expected to increase galaxy size more than they do stellar mass, the most direct way to test this is to measure the stellar mass--size relations in both the cluster and field environments at fixed redshift and compare them to see if there is a significant offset in size. If the predictions of minor mergers driving galaxy size growth are true, cluster galaxies should find themselves inhibited from size growth and will therefore be significantly smaller than field galaxies at fixed stellar mass. In Chapter 2 of this thesis, we do this experiment at z∼1, finding that quiescent cluster galaxies are smaller than quiescent field galaxies at fixed stellar mass. This supports the case for minor mergers driving size growth in quiescent field galaxies.

Nevertheless, the process whereby large star-forming galaxies quench and join the quiescent population at the large size end has also been suggested as an explanation for the size growth of quiescent galaxies. Using ancillary spectroscopy of our 10 clusters from the Gemini Cluster Astrophysics Spectroscopic Survey (GCLASS), we pick out 23 spectroscopically confirmed recently quenched galaxies in the clusters and study their position on the stellar mass--size relation in Chapter 3. We find that they follow a mass--size relation lying midway between the star-forming and quiescent relations. This result provides direct evidence showing galaxies which quench later are on average larger than the bulk of the quiescent galaxy population at fixed stellar mass and redshift. This work showed that at least in the cluster environment, recently quenched galaxies will induce a rise in the average size of quiescent galaxies with decreasing redshift.

Finally, this thesis attempts to tackle one of the biggest unanswered questions in galaxy evolution: how does quenching operate in the high-redshift Universe? Surveys such as GASP and VESTIGE have already allowed us to build a comprehensive understanding of environmental quenching in the local Universe. Obtained using the WFC3 G141 grism, we use spatially resolved Hα maps of cluster and field galaxies at z∼1 to directly observe where star formation is occurring in these galaxies and where it is not. In Chapter 4, we measure the stellar mass--size relations of z∼1 star-forming cluster and field galaxies in F140W and Hα. The lack of a clear environmental quenching signature in this work hints at the rapidity of environmental quenching in the high-redshift Universe, and/or a complete change in the physics of environmental quenching.