LCFRI · Connections between algorithmic and geometric properties of groups

Connections between algorithmic and geometric properties of groups

Summary overview of results The research we proposed lies at the intersection of group theory, theoretical computer science, logic and geometry, with a particular emphasis on the study of equations in groups, and related decision problem in groups. The goal of the project was to deduce information about the decidability and complexity of some important algorithmic questions about groups based on the geometric properties displayed by the groups. Most of the work performed during the first half of the grant has been concerned with the solvability and complexity of some natural decision problems in groups. I have studied the endomorphism and monomorphism problems, and more generally investigated computational, combinatorial and asymptotic issues around the solvability of equations in groups, with or without particular logical constraints. During the second part of the grant my work has turned from the decision problems from an algorithmic point of view towards the languages that describe them, and has led to results about geodesics, conjugacy representatives and geodesics, and their growth, and applications of geodesics'factorisations to the analytical side of group theory. See http://homeweb1. unifr. ch/ciobanul/pub/GroupTheory/for a complete list of publications that resulted from the work done during the grant. -Conclusions and the socio-economic impact of the project The work we have done during the Marie Curie Reintegration grant has strengthened the connections between the algorithmic side of group theory and the geometric and analytic side. We have invited a large number of mathematicians from Europe, USA and Australia to Fribourg, and have participated in many international conferences, as well as presented work in seminars in Switzerland and abroad. This has created active collaborations with group theorists, theoretical computer scientists and logicians, and has added to the outreach of one of the most exciting area in mathematics nowadays, geometric group theory. I want to reemphasize in this section my extremely positive experience with the grant. There are some important factors that have made this grant be significant for the development of my career. The first factor I would like to mention is that it allowed a young researcher (me) to attract an impressive number of visitors and collaborators during a very sensitive time, when the researcher was making the transition between postdoctoral positions, and building up a resume towards a permanent position in the academia. Often the grants in science go to the most senior researcher and the most famous institution. This Marie Curie Reintegration grant went to a relatively unknown scientist in a small institution, and it thus had a tremendous impact and visibility. It also allowed me to organise the first international conferences in my career (together with Tatiana Smirnova from Geneva), and make myself visible to the wider scientific community. As an organiser, I have been very sensitive to the social composition of the speakers and participants, using the funds in the Marie Curie grant to invite women, young PhD students or postdocs, as well as participants from countries with more limited funding. This somewhat reflects the characteristics of my background: I am a young woman coming from an Eastern European country. Finally, I would like to mention the importance this grant had in sustaining my research activity during the pregnancy and the subsequent time when I was the main care giver for my baby. The funding provided by the Marie Curie grant has allowed me to have guests at a time when it was impossible for me to travel. This made the grant more important to me than I initially imagined before I had a child. My experience with the Marie Curie grant during motherhood has thus been extremely positive. This grant, together with the support of my host institution, University of Fribourg, and the scientist in charge, Prof. Ruth Kellerhals, has made it possible for me to be active scientifically during a very challenging time

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