Final Report Summary - MODGROUP (The Model Theory of Groups)
The Intra-European Fellowship MODGROUP (2010--2012) funded a fellowship by Dr. J. Gismatullin, hosted at the University of Leeds with Prof. H.D. Macpherson as Scientist-in-Charge. The research area was model theory, a branch of mathematical logic which investigates mathematical structures from the point of view of first order logic, in particular the 'definable sets' in such structures (solution sets of first order formulas). In the project, model theory was applied to group theory, a central branch of algebra, and the focus was mainly on groups for which the model theory is 'tame', that is, definable sets can be understood. However, some aspects of the project were purely group-theoretic, though the questions asked all had model-theoretic motivation.
The objectives of the project concerned various 'connected components' in a group G, assumed to be definable in a 'sufficiently saturated' structure M. These connected components are denoted G(0) (the largest), G(00), and G(000) (the smallest), and are all normal in G. They need not always exist, but Gismatullin mainly worked under assumptions which guarantee their existence (e.g. that the first order theory of M has the property 'NIP'). The corresponding quotient groups G/G(0), G/G(00) and G/G(000) are quasi-compact topological groups which are invariants of the underlying first order theory, and have interest for varied reasons, both model-theoretic and group-theoretic -- for example in o-minimal settings the quotient G/G(00) is a Lie group and the corresponding quotient map is analogous to a standard-part map in non-standard analysis.
The objectives of the project mainly concerned the structure of the quotient group G(00)/G(000), and in particular the question whether it is always trivial. There were subsidiary questions, motivated by this, concerning Gismatullin's notion 'absolute connectedness' of a group (in particular, for algebraic groups), and concerning 'bounded simplicity' (a strong form of group-theoretic simplicity which is first order expressible and implies absolute connectedness).
At an early stage of the project, Conversano and Pillay (in Leeds) gave an example of a group G in an o-minimal structure M for which G(00) and G(000) are distinct. The group G is essentially a universal cover of SL(2,R). This answered some questions in the project, but opened up many others, on which Gismatullin then focused. In two papers he studied the quotient group G(00)/G(000). In one (currently submitted, with Krupinski), he gave new examples where this quotient is non-trivial, by identifying a certain condition on 2-cocycles which guarantees this, exploiting the Matsumoto-Moore theory around 2-cocycles. In another paper (submitted) he gave an example where the quotient is non-abelian (a question asked by Pillay), and introduced a new notion of 'generalised quasimorphism' and a method for recovering a compact Hausdorff group from a dense subgroup. In the same paper he investigated absolute connectedness in the very important class of Chevalley groups (over arbitrary fields), essentially proving absolute connectedness and thereby proving Conjecture 3 of the proposal. He also showed that the class of 'type absolutely connected' groups is exactly that of discretely topologised groups with 'trivial Bohr compactification', thereby discovering and exploiting an interesting connection to another subject area.
The notion of bounded simplicity was investigated by Gismatullin in two further papers. In one (submitted) he extends very important classical work on simplicity of certain groups of automorphisms of trees, characterising when such groups are boundedly simple. In the other (still in preparation) he investigates bounded simplicity in certain `isotropic' groups related to simple algebraic groups -- these are groups of rational points over arbitrary fields.
Gismatullin wrote with Chen (a PhD student in Leeds) and Bowler (then a postdoctoral fellow in Cambridge) a significant paper (to appear in the Journal of Symbolic Logic) on connected components for finitely generated nilpotent groups. This work has a very different flavour to that described above, and is much closer to additive combinatorics. In particular, they give a group-theoretic characterisation of G(0), and investigate the model theory of expansions of an infinite cyclic group.
Partly due to the work of Conversano and Pillay mentioned above, Gismatullin began a collaboration (also related to connected components) with Pillay and Penazzi (a postdoctoral fellow in Leeds) which makes a new connection between model theory and topological dynamics. One paper has been submitted, and another is in preparation. Topological dynamics is concerned with 'G-flows', that is, continuous actions of topological groups G on compact spaces. If G is a definable group there is a natural 'definable' G-flow, namely the action on the space of global types of elements of G. In one paper, they describe a universal minimal G-flow for SL(2,R) in this setting, and in another, they develop the general model theory of G-flows for topological groups. In the first paper they also answer negatively an interesting question of Newelski.
Gismatullin has a number of other papers in preparation. In joint work with Chen, he answers an old group-theoretic question of Stallings, exploiting recent work of Nikolov and Segal on finite groups. This work has the possibility of very strong applications to the theory of sofic groups. In another paper, he extends an old method of Lascar to prove bounded simplicity of certain automorphism groups of field extensions. In joint work with Macpherson and Simonetta, he examines the class of groups first order definable in an algebraically closed valued field, characterising the case when the group is simple (at least, under an extra hypothesis). This work may be considerably extended.
So far, one paper has been accepted for publication and four are currently submitted to journals, and a further five are in preparation. In addition, Gismatullin gave a number of invited talks in seminars and conferences, and attended a large number of other conferences. He paid a number of research visits to other institutions to develop collaborations, and built a number of new contacts.
Gismatullin's presence in the logic research group of the University of Leeds was extremely valuable. He provided stimulus and inspiration for Macpherson and Pillay and also collaborated with a PhD student and a postdoctoral fellow. This collaboration will have been particularly valuable for the PhD student. He participated in 3--4 weekly seminar series, sometimes giving talks, and always showing active interest and asking questions. In this way, and also through some complementary training, he took strong advantage of training opportunities in Leeds afforded by the project.
Overall, the project produced important research on the model theory/ group theory borderline, helped position Gismatullin for a research career in Europe, and provided stimulus for the research group in Leeds.
The objectives of the project concerned various 'connected components' in a group G, assumed to be definable in a 'sufficiently saturated' structure M. These connected components are denoted G(0) (the largest), G(00), and G(000) (the smallest), and are all normal in G. They need not always exist, but Gismatullin mainly worked under assumptions which guarantee their existence (e.g. that the first order theory of M has the property 'NIP'). The corresponding quotient groups G/G(0), G/G(00) and G/G(000) are quasi-compact topological groups which are invariants of the underlying first order theory, and have interest for varied reasons, both model-theoretic and group-theoretic -- for example in o-minimal settings the quotient G/G(00) is a Lie group and the corresponding quotient map is analogous to a standard-part map in non-standard analysis.
The objectives of the project mainly concerned the structure of the quotient group G(00)/G(000), and in particular the question whether it is always trivial. There were subsidiary questions, motivated by this, concerning Gismatullin's notion 'absolute connectedness' of a group (in particular, for algebraic groups), and concerning 'bounded simplicity' (a strong form of group-theoretic simplicity which is first order expressible and implies absolute connectedness).
At an early stage of the project, Conversano and Pillay (in Leeds) gave an example of a group G in an o-minimal structure M for which G(00) and G(000) are distinct. The group G is essentially a universal cover of SL(2,R). This answered some questions in the project, but opened up many others, on which Gismatullin then focused. In two papers he studied the quotient group G(00)/G(000). In one (currently submitted, with Krupinski), he gave new examples where this quotient is non-trivial, by identifying a certain condition on 2-cocycles which guarantees this, exploiting the Matsumoto-Moore theory around 2-cocycles. In another paper (submitted) he gave an example where the quotient is non-abelian (a question asked by Pillay), and introduced a new notion of 'generalised quasimorphism' and a method for recovering a compact Hausdorff group from a dense subgroup. In the same paper he investigated absolute connectedness in the very important class of Chevalley groups (over arbitrary fields), essentially proving absolute connectedness and thereby proving Conjecture 3 of the proposal. He also showed that the class of 'type absolutely connected' groups is exactly that of discretely topologised groups with 'trivial Bohr compactification', thereby discovering and exploiting an interesting connection to another subject area.
The notion of bounded simplicity was investigated by Gismatullin in two further papers. In one (submitted) he extends very important classical work on simplicity of certain groups of automorphisms of trees, characterising when such groups are boundedly simple. In the other (still in preparation) he investigates bounded simplicity in certain `isotropic' groups related to simple algebraic groups -- these are groups of rational points over arbitrary fields.
Gismatullin wrote with Chen (a PhD student in Leeds) and Bowler (then a postdoctoral fellow in Cambridge) a significant paper (to appear in the Journal of Symbolic Logic) on connected components for finitely generated nilpotent groups. This work has a very different flavour to that described above, and is much closer to additive combinatorics. In particular, they give a group-theoretic characterisation of G(0), and investigate the model theory of expansions of an infinite cyclic group.
Partly due to the work of Conversano and Pillay mentioned above, Gismatullin began a collaboration (also related to connected components) with Pillay and Penazzi (a postdoctoral fellow in Leeds) which makes a new connection between model theory and topological dynamics. One paper has been submitted, and another is in preparation. Topological dynamics is concerned with 'G-flows', that is, continuous actions of topological groups G on compact spaces. If G is a definable group there is a natural 'definable' G-flow, namely the action on the space of global types of elements of G. In one paper, they describe a universal minimal G-flow for SL(2,R) in this setting, and in another, they develop the general model theory of G-flows for topological groups. In the first paper they also answer negatively an interesting question of Newelski.
Gismatullin has a number of other papers in preparation. In joint work with Chen, he answers an old group-theoretic question of Stallings, exploiting recent work of Nikolov and Segal on finite groups. This work has the possibility of very strong applications to the theory of sofic groups. In another paper, he extends an old method of Lascar to prove bounded simplicity of certain automorphism groups of field extensions. In joint work with Macpherson and Simonetta, he examines the class of groups first order definable in an algebraically closed valued field, characterising the case when the group is simple (at least, under an extra hypothesis). This work may be considerably extended.
So far, one paper has been accepted for publication and four are currently submitted to journals, and a further five are in preparation. In addition, Gismatullin gave a number of invited talks in seminars and conferences, and attended a large number of other conferences. He paid a number of research visits to other institutions to develop collaborations, and built a number of new contacts.
Gismatullin's presence in the logic research group of the University of Leeds was extremely valuable. He provided stimulus and inspiration for Macpherson and Pillay and also collaborated with a PhD student and a postdoctoral fellow. This collaboration will have been particularly valuable for the PhD student. He participated in 3--4 weekly seminar series, sometimes giving talks, and always showing active interest and asking questions. In this way, and also through some complementary training, he took strong advantage of training opportunities in Leeds afforded by the project.
Overall, the project produced important research on the model theory/ group theory borderline, helped position Gismatullin for a research career in Europe, and provided stimulus for the research group in Leeds.