Generative Artificial Intelligence (GenAI) is revolutionizing higher education, offering new opportunities for personalized learning, assessment and competency development. This paper describes the creation of a learning community (LC) on GenAI at the University of Alicante. This community, designed for the Teaching and Research Staff (PDI) and the Technical, Management, and Administrative Staff (PTGAS), aims to share knowledge and foster collaboration on GenAI applications in teaching/learning processes and in the procedures and tasks carried out by PTGAS. Due to its accessibility and features, Moodle has been chosen to host the LC. The structure includes forums, materials, audiovisual resources, and synchronous meetings. Initial results indicate significant engagement with the content but limited participation in the forums. Following the announcement of the first in-person workshop, a notable increase in community members was observed, suggesting a preference for traditional learning formats. These preliminary findings highlight the challenges in promoting horizontal communication and suggest the need for additional engagement strategies. The ongoing development of this LC aims to enhance user participation and fully integrate GenAI into teaching/learning processes and university management. This study provides some preliminary conclusions on the adoption of GenAI in higher education, emphasizing the importance of adaptive strategies to foster effective LC.