# Jose Maria Montesinos Amilibia on Francisco González Acuña

We give an English translation of J M Montesinos Amilibia, 'Francisco Gonzalez Acuña: Semblanza Humana e Intelectual',

*Centro de Ciencias Matemáticas, UNAM*(September 2002).**Francisco Gonzalez Acuña: Human and Intellectual Profile.**

It is my turn today to carry out a very pleasant task. It is about writing a few words in honour of a very dear friend and collaborator, an illustrious mathematician and one of the most penetrating and luminous intelligences that the great Nation of Mexico has today. I am very grateful for the honour that the Mexican Mathematical Society gives me, through its editors, by asking me to write this brief note. And I do it from the bottom of my heart, even though at the very moment I write, I have the inability of my poor pen to express even weakly the feelings of appreciation and respect that I harbour towards that phenomenon of mathematics that we call Fico.

I was lucky to learn of Fico's existence thanks to a conversation with Ignacio Canals, who at the time, in 1972 I believe, was visiting Francisco Botella in Madrid, "my professor" (as it was said then) who was from the Chair of Analytical Geometry and Topology at the Complutense University of Madrid. I had just written an article, in Spanish, and wanted to send it "outside." Professor Canals told me with his soft Mexican accent: "why don't you send it to the Bulletin, in Mexico; Fico will surely see it there; and I will submit it for you." And thus began an epistolary relationship that resulted in an article written in three parts (with J Birman) without either knowing the other personally. When exchanging letters in connection with this work, I realised that I was dealing with an unusual mathematician: his interest in generalising to limits inconceivable to me filled me with astonishment, no less than did his peculiar way of writing. It was clearly seen, in that child's primer paper and in the firm but irregular strokes of a blunt pencil, and moistened with bites, that this was the work of a medieval miniaturist. These first contacts were so fruitful that I was itching to see the flesh and blood Fico face to face.

And the occasion presented itself in the summer of 1977. Fico invited me to the Institute of Mathematics, he met me at the airport in his nondescript car, and we spent three months working together. This is one of the experiences that every time it is repeated, on the occasion of our mutual encounters, gives me greater pleasure. Anyone who has been lucky enough to talk to Fico about mathematics knows this. We talked about it in the car; on the street; at the dentist's surgery; buying tortillas; playing baseball; in his house, surrounded by children and with his better half, the ineffable Susana, who, I remember, made all kinds of extra-mathematical comments. Finally, everywhere. But Fico did not lose his temper or the thread. It was patience in person. And from one day to the next, he continued to keep his pulse on whatever the problem was. And another thing that filled me with admiration was his phrase: "You're right!" As if the happy idea had come from me! A simple observation that occurred to me on the fly plunged him into a calm silence that after a long time crystallised into his usual "You're right!". I have to confess that he almost always caught me on the wrong foot: he had been thinking for such a long time that my mind had already forgotten what I had told him, and when he repeated it to me, I did not recognise it: it was almost always an impeccable mathematical statement. I know that some will say that I exaggerate; but I do not; it was in those months when my respect and admiration for such penetrating intelligence, together with such good manhood, such generosity and such a fine sense of humour, forged the unbreakable friendship that unites me with Fico.

Another of Fico's characteristics, which I believe has a profound impact on his way of viewing mathematics, is his passion for the game. Any game, in his hands, becomes mathematics. And I'm about to say that he conceives of mathematics as a game. Characteristic of the game is the aspect that it has as a challenge, as a "little bull" as he would say. And so many of his most important mathematical results solve more or less famous questions. For example, Neuwirth's U problem (in his book

*Knot Groups*) about whether a finitely generated group, with deficiency less than or equal to zero and weight one, is a homomorphic image of the group of a knot, is solved by Fico in a beautiful and terse article in the

*Annals of Mathematics*in which some observations by J Milnor are also reviewed. Another case, of which I was a close witness, is what Fico called (I don't know if it was a joke) "Fox's last problem": about whether there are groups of 2-nodes with infinite terminals, also solved and published in the same journal, and that likewise contains some suggestions of J Milnor.

Obviously, this is not the place (nor the tone) to do justice to Fico's mathematical work. From his first works, in the

*Annals*of the Institute of Mathematics (UNAM), one can already see the unmistakable style that will follow him later to his most recent publications: love of precision, intelligent generalisation, the chess strategy of checkmate. His work is perfect. Today, when so many poorly written and almost incomprehensible works appear, Fico's style shines like a beacon in the dark. His work is relatively extensive and all of it is excellent:

*pauca sed matura*seems to be his motto. Witnesses: the numerous manuscripts that are retained unpublished in the bottom of his inseparable bag. I have seen appearing from his bag, the same manuscript that I saw twenty years ago, retouched a thousand times. But that's Fico. Quiet man, enemy of fuss, and limitless tenacity.

Fico was, like Milnor and many others that we cannot remember here, a disciple of Ralph H Fox. Under the guidance of this great mathematician from Princeton University, Fico completed his doctoral thesis in 1969, of which I fondly keep a copy. It was published in part in volume 15 of the year 1970 of the Bulletin of the Mexican Mathematical Society under the title "Dehn's construction on Knots". This work is a true geometric mine, which will mark the lines of force of Fico's future research. There many of the problems that he will later solve alone or with the help of one of his multiple collaborators appear in germ. Well, a very characteristic of his is the ease of always putting himself at the level of knowledge of his interlocutor. That, which is the most valuable garment of the good teacher, is instinctive in him. Quietly, slowly, he develops his ideas so that following him is easy and pleasant. Collaboration with him is therefore natural and stimulating. That is why his lectures are so appreciated: good style; good mathematics. As L Neuwirth says in the introduction to the book

*Knots, groups and 3-manifolds*, published in memory of Ralph H Fox: "The influence of a great teacher and superb mathematician is measured by his published work, by the published works of his disciples, and, perhaps most important, for the mathematical atmosphere it created and helped to maintain." I think that in these three facets Fico's work is equally memorable. May we continue to enjoy his friendship, his fine humour, his joy and his mathematics. Fico: congratulations on your 60th "twist spinning"!

Last Updated June 2023