**Fritz Grunewald**'s parents were Helene Weisbrod and Friedrich Grunewald. Friedrich, a dentist and excellent sportsman, married Helene, who was brought up on a farm in Lambsheim, and the couple settled in Bad Kreuznach. Helene and Friedrich Grunewald only had one child, Fritz (the subject of this biography), and they separated when he was only two years old. Helene and Fritz then went to live on the family farm at Lambsheim where the boy was brought up. It was not an easy childhood, however, and the strains in the family are indicated in [2]:-

Helene remarried quite soon, remaining on the farm; Fritz's stepfather Kurt Kinkel was[like Helene and her mother]a rather dominating personality, and relations between him and the young Fritz were difficult. Kinkel disapproved of Grunewald senior, while Fritz naturally wanted to maintain contact with his father. He felt the resulting tension throughout his childhood; by the time he was sixteen the tension proved excessive and Fritz and his father broke off all contact. By this time Grunewald senior had remarried; his daughter Ulrike has fond memories of her older half-brother, and recalls the pain of losing touch with him when she was only seven. It took twenty years for Fritz to re-establish contact with his father and with Ulrike; the long-separated siblings became close friends.

Fritz attended the Albert-Einstein-Gymnasium in Frankenthal, about 8 km east of Lambsheim, and graduated in 1968. An outstanding pupil, Fritz obtained the best marks in his Arbitur of any of his classmates. He had been particularly fond of mathematics throughout his years at the Gymnasium so he matriculated at Göttingen University in 1969 to begin his university study of mathematics and physics. One of his lecturers was the group theorist Jens Mennicke who soon gave Grunewald a love of group theory; it became a topic on which he would undertake research throughout his life. Mennicke left Göttingen when he was appointed to a chair at Bielefeld University in 1971 and Grunewald followed him there to complete his undergraduate studies. At Bielefeld he studied for his Diploma supervised by Andreas Dress and, in 1972, was awarded a Diploma having written a dissertation on Hecke rings of finite groups. He continued to study for his doctorate at Bielefeld, advised by Jens Mennicke, and he was awarded the degree in 1974 for his thesis *Über eine Gruppe vom Exponenten acht* Ⓣ.

In his doctoral thesis Grunewald studied the Burnside group with two generators and exponent 8. It was an open question whether this group, denoted by *B*(2,8), was finite or infinite. Although Grunewald failed to answer this question, he did show that a group of exponent 8 that is generated by elements *x*, *y* such that *x*^{2} = *y*^{4} = (*xyxy*^{2})^{4} = 1 is finite. When he submitted this result in a paper coauthored with Jens Mennicke, for publication in a journal, the referee commented that:-

... this belongs in the Guinness book of records, not a mathematical journal!

The question concerning the finiteness of *B*(2,8) has proved exceedingly hard and remains open today. In fact, perhaps hoping for even further progress, it was some years before Grunewald published papers based on the results he had obtained in his doctoral thesis. After his doctoral studies, Grunewald spent the year at 1973-74 at Queen Mary College, London. There he began a collaboration with Dan Segal (the author of [2], [3] and [4]) and this led to Grunewald's first publication, namely the paper *Residual nilpotence in polycyclic groups* (1975) written jointly with Segal. In this paper they showed that a polycyclic group with the property that each of its 2-generator subgroups is residually nilpotent, need not necessarily be itself residually nilpotent.

Grunewald had met Barbara Gernhuber (born 1951 in Bonn) when she was studying law in Bielefeld. Barbara had graduated from the Uhland-Gymnasium in Tübingen in 1970 and, after studying at Bielefeld University joined Grunewald during his year in London. Barbara and Fritz married in 1974 and continued their studies at Bielefeld. where Grunewald was appointed as an assistant. Their first two children, Natalie (born 1976) and Joachim (born 1978), were born in Bielefeld. Grunewald habilitated at Bielefeld in 1979 and his wife Barbara was awarded her doctorate by Bielefeld in the following year for her thesis* Grenzziehung zwischen der Rechts- und der Sachmängelhaftung beim Kauf* Ⓣ.

The award of a prestigious Heisenberg scholarship allowed Grunewald to move with his family to the University of Bonn. In 1981 he was appointed as an assistant professor at Bonn. Segal writes [2]:-

Fritz and Barbara had a third child, Andreas, in1986. Barbara's distinguished career as a law professor ran alongside his own, and Fritz always took his share of domestic and parental responsibilities, arranging his work around the imperatives of childcare, shopping and cooking. In those days this was unusual among(male)German professors, and occasionally led to raised eyebrows among some colleagues in the department; but his priorities were clear.

In fact, after the birth of their third child, Barbara habilitated at the University of Bonn in 1987 with her thesis *Ausschluss aus Gesellschaft und Verein* Ⓣ. In 1992 Grunewald made his final career move when he was appointed to the chair of mathematics at the Heinrich-Heine University of Düsseldorf.

Dan Segal writes [4]:-

[

Grunewald's]numerous important contributions range ... over infinite group theory, finite group theory, Diophantine decision problems, arithmetic groups, automorphic forms and algebraic geometry. A widely influential figure, he was particularly inspiring as a collaborator.

Grunewald wrote the important book *Groups Acting on Hyperbolic Space*(1998) in collaboration with Jürgen Elstrodt and Jens Mennicke. Stefan Kühnlein begins a review by writing that the book:-

... is a very welcome contribution to harmonic analysis of hyperbolic three-manifolds and the related number theory. It is the first time that most of the relevant results in this area have been brought together in such a coherent way. The book is written very carefully, with much expertise, and the authors have taken every effort to provide as clear an exposition as possible. All chapters, and many sections, begin with a description of the purpose they serve. Chapters are concluded with helpful remarks on omissions and history. Remarks of this kind are also to be found throughout the text. The authors treat the case of dimension three only, and there truly is enough to say about this. The mathematical language is a classical one, and representation theory is not used. In this respect, the book is a complement to much of the recent literature on automorphic forms.

Kühnlein ends his review with high praise:-

... the book under review is a brilliant introduction for non-experts to its subject. For the expert it will be a major source of inspiration and indispensable for reference. It should be part of every mathematical library, though the reviewer strongly recommends not leaving it there.

Grunewald died from a sudden heart attack one week before his 61^{st} birthday. His 60^{th} birthday had been celebrated a year earlier with a well-attended conference 'Group theory, number theory and geometry' held at the University of Oxford. A special issue of the journal *Groups, Geometry, and Dynamics*, to mark his 60^{th} birthday was edited by Martin Bridson and Dan Segal with the cooperation of Alex Lubotzky and Peter Sarnak. Sadly Grunewald died before the special issue appeared as Volume 5, Part 2, in 2011.

Two honours which Grunewald received should be mentioned. These are the Reinhard and Emm Heynen Prize from the University of Düsseldorf in 2001, and the invitation to lecture at the International Congress of Mathematicians held in Madrid in 2006.

**Article by:** *J J O'Connor* and *E F Robertson*