Celebratio Mathematica

Low-di­men­sion­al geo­metry and to­po­logy has been a won­der­ful area to ex­plore, with many great con­trib­ut­ors. Peter Scott has been a lead­er and has shared his ideas and in­sights gen­er­ously with col­lab­or­at­ors. I have been very for­tu­nate to work with Peter and learn a lot from him. In par­tic­u­lar, his ideas about ends of pairs of groups and the con­nec­tions to the sin­gu­lar­it­ies and com­bin­at­or­ics of least area sur­faces have been a big in­flu­ence on me.

I will al­ways re­mem­ber Peter ex­plain­ing to me how the right way of think­ing about count­ing double curves for im­mersed in­com­press­ible sur­faces is to work in the cov­er­ing space cor­res­pond­ing to the fun­da­ment­al group of such a sur­face. This cov­er­ing space is eas­ily seen to be ho­mo­topy equi­val­ent to a product of the sur­face and a line, and the sur­face lifts to this cov­er­ing space (in case the sur­face is not a vir­tu­al fibre) are then a single copy of the sur­face and vari­ous non­com­pact sur­faces. These lat­ter sur­faces have ends at the ends of the ho­mo­topy cyl­in­der cov­er­ing space. The right way to count the double curves is one for each such non­com­pact lift hav­ing at least one end at each end of the ho­mo­topy cyl­in­der. (For sim­pli­city, I am as­sum­ing both the sur­face and am­bi­ent 3-man­i­fold are ori­ent­able). This count clearly only de­pends on the ho­mo­topy class of the sur­face im­mer­sion, not on the map­ping chosen to rep­res­ent it. The ac­tu­al num­ber of double curves can vary de­pend­ing on the map­ping choice but not this ver­sion, based on the po­s­i­tion of ends of the sur­face lifts.

For im­mersed geodesics on a hy­per­bol­ic sur­face, there is a sim­il­ar way of count­ing double points us­ing an ap­pro­pri­ate cyc­lic cov­er­ing space. In this case, the count based on ends does give the num­ber of double points.

Peter writes beau­ti­ful pa­pers, full of ideas and with great clar­ity. His re­view art­icle on the geo­met­ries of 3-man­i­folds has been read and ab­sorbed by le­gions of gradu­ate stu­dents and young re­search­ers in­ter­ested in un­der­stand­ing Thur­ston’s geo­met­riz­a­tion pro­gram. His pa­per that there are no exot­ic Seifert fibred spaces with in­fin­ite fun­da­ment­al group1 and the pa­per with Bill Meeks on group ac­tions2 neatly tie to­geth­er cov­er­ing space the­ory and the power of least area in­com­press­ible sur­faces.

My fi­nal com­ment is about the joy and en­thu­si­asm of talk­ing about math­em­at­ics with Peter. This has been a great pleas­ure for me.