# Permutation Groups (16w5087)

Arriving in Banff, Alberta Sunday, November 13 and departing Friday November 18, 2016

## Organizers

George Willis (The University of Newcastle)

Cheryl Praeger (The University of Western Australia)

Katrin Tent (Universitaet Muenster)

Donna Testerman (Ecole Polytechnique Federale de Lausanne)

## Objectives

We propose a meeting focusing on these four main themes.

Finite permutation groups, graphs, geometries, and simple groups

Very recent results about finite permutation groups (many still unpublished) suggest that the finite simple groups still hold secrets about simply stated properties of finite permutation groups and the structures on which they act: for example, in most primitive permutation groups each element has a regular cycle (of maximum possible length, equal to the element order), and in most primitive groups each element has order less than a quarter of the number of points; and in both cases we can describe the exceptions. The young researchers involved in these results are among our list of participants.

New studies of symmetric structures demand strong new theoretical tools from permutation groups. The question of base size of permutation actions is of importance in computational group theory as well as in the study of the graph isomorphism problem. Recent research (by Burness, Fawcett, and more senior players) has thrown much light on the base sizes of actions of almost simple groups in particular -- to such an extent that the major obstacle to proving Pyber's base size conjecture is a sufficiently detailed understanding of finite imprimitive linear groups. Similarly Li's studies of edge-transitive embeddings of vertex-primitive graphs in compact Reimann surfaces leads to new factorisation problems for finite almost simple groups.

From the opposite point of view, breakthroughs in combinatorics and geometric group theory lend new methods for studying group actions: for instance, the exciting results of Helfgott on expansion in groups, extended recently by Pyber and Szab\'o, and independently by Breuillard, Green and Tao, have shown not only that Cayley graphs of bounded rank finite simple Lie type groups are expanders, but have also yielded a proof of the Weiss Conjecture of 1978 for locally primitive graphs involving only bounded rank composition factors. This highlights the classical Lie type groups of unbounded rank as those for which greater understanding is needed.

We believe that this workshop will spark substantial progress on some of these questions, and the workshop will provide a perfect opportunity to bring together senior researchers in this theme (such as Cameron, Guralnick, Li and Tiep) with young contributors in the area (Fawcett, Giudici, Guest, Morgan, Spiga, Verret). In addition these researchers work across the diferent foci of the workshop.

Infinite permutation groups and model theory

Over the last years there have also been major developments in infinite permutation group theory. One aspect here is the interaction between permutation group theory, combinatorics, model theory, and descriptive set theory, typically in the investigation of first order relational structures with rich automorphism groups. The connections between these fields are seen most clearly for permutation groups on countably infinite sets which are closed (in the topology of pointwise convergence) and oligomorphic (that is, have finitely many orbits on $k$-tuples for all $k$); these are exactly the automorphism groups of $\omega$-categorical structures, that is, first order structures determined up to isomorphism (among countable structures) by their first order theory.

Themes of current activity here include the following.

(a) The use of group theoretic means (O'Nan-Scott, Aschbacher's description of maximal subgroups of classical groups, representation theory) to obtain structural results for model-theoretically important classes (totally categorical structures, or much more generally, smoothly approximable structures, finite covers of well-understood structures). Recent progress by Macpherson, Kaplan and Simon towards showing that natural classes of automorphism groups are maximal closed subgroups of the symmetric group give rise to hope for a complete classification in certain settings.

(b) Infinite $2$-transitive groups are far from being classified, even in the split or the sharply $2$-transitive case. Here model theory comes in on the one hand as a means towards constructing examples as well as a setting of special cases like the assumption of finite Morley rank where such a classification would be essential. Recently the first non-split sharply 2-transitive groups were constructed by Rips, Segev and Tent. The existence of non-split sharply $2$-transitive groups is still open in all characteristics different from 2 and 3, but recent progress should be discussed to obtain hopefully a complete picture.

(c) We plan to further investigate properties which the full symmetric group $S$ on a countable set shares with various other closed oligomorphic groups like uncountable cofinality, the existence of conjugacy classes that are dense or even comeager in the automorphism group, or the Bergman property. Model theoretic methods have recently been used to obtain a general criterion for the abstract simplicity of such automorphism groups or in other cases to determine the full normal subgroup structure. The criterion applies to diverse examples such as the Urysohn space, random graphs, random hypergraphs, or random tournaments.

Totally disconnected locally compact groups and permutation groups

Infinite permutation groups with finite sub-degrees provide an approach to studying totally disconnected, locally compact groups (t.d.l.c groups). On one hand, every such group has associated infinite graphs, known as the `rough Cayley', `Schreier' or `Abels-Cayley' graphs, on which it acts transitively with finite sub-degrees while, on the other, every transitive group action on an infinite set having finite sub-degrees gives rise to a totally disconnected, locally compact group. New concepts such as the scale, tidy subgroups and flatness have been introduced in the structure theory of totally disconnected, locally compact groups and a typology for simple groups is being developed. This progress is leading to an exchange of ideas and problems with the theory of infinite permutation groups.

Primitivity of infinite permutation groups, as it applies to totally disconnected groups, has been studied in pioneering work by S. Smith on orbital digraphs and subdegree growth. Finiteness of sub-degrees in permutation representations corresponds to the structural fact that t.d.l.c. groups contain open profinite subgroups and the values of these sub-degrees correlate with values of the scale. Work on commensurators of profinite groups by Barnea, Ershov and Weigel initiated a study of how their profinite local structure influences the global structure of t.d.l.c. groups, which is being taken further by Caprace, Reid and Willis in work that bases a typology of simple t.d.l.c. groups on this local structure.

When the infinite set being permuted comprises the vertices, or other objects, in a geometry, the geometrical intuition resulting is a powerful method for understanding the group. For example, the flat-rank of Kac-Moody t.d.l.c. groups corresponds to the dimension of flats in their associated building. Automorphism groups of trees and buildings are, through the work of Tits, Haglund and Paulin, a rich source of t.d.l.c. groups. This work has inspired attempts at general constructions of canonical geometries for t.d.l.c. groups by Baumgartner, Ramagge and Willis, with one outcome being a `space of directions' in which it is possible to move to infinity within the group.

A link between actions on graphs and the scale and tidy subgroups was made early on by M\"oller when he characterised tidiness of a subgroup in terms of a certain graph being a regular rooted tree whose valency is the scale. A higher rank version of this, that characterises flatness in terms of the structure of a graph generalising the one considered by M\"oller, would give another translation between group structure and actions on graphs.

Algebraic groups, subgroup structure and permutation group actions The large majority of the finite quasisimple groups are finite groups of Lie type, which are obtained as fixed point subgroups of certain endomorphisms of reductive algebraic groups. A main aim is to understand the subgroup lattice of these groups, knowledge of which is required for the study of permutation actions of the groups. Well-known reduction theorems motivate the study of so-called irreducible triples $(X,Y,V)$, where $X$ is a subgroup of $Y$ and $V$ is an absolutely irreducible module for $Y$ on which $X$ acts irreducibly. Recent progress on the classification of such triples has been achieved in the work of Husen, Hiss and Magaard on imprimitive actions for finite groups, Burness, Ghandour, Marion and Testerman, on irreducible triples for classical algebraic groups. In addition, many people are actively contributing to the study of irreducible triples in the finite groups. In a different direction, the recent work of Litterick, Stewart, and Thomas aims at understanding the subgroup lattice of an exceptional type algebraic group $G$. They consider $G$-irreducible subgroups, and $G$-non completely reducible subgroups, as well as so-called Lie-primitive finite subgroups, work leading to whole new insights on the subgroup stucture of the exceptional groups.

In another direction, Burness, Guralnick and Saxl have undertaken an extensive study of the problem of base size for algebraic group actions, obtaining the precise value for almost all primitive actions of simple algebraic groups, and applying their results to the study of the essential dimension of simple algebraic groups. A related question, studied in on-going work of Guralnick, Lawther, Liebeck, and Martin, is that of regular orbits in algebraic group actions. They determine for which irreducible modules $V$ for a simple algebraic group, the generic stabilizer is non trivial. That is, when does there exist a non-empty open subset of $V$ such that for all points in the set the stabilizer is non trivial. There are new and interesting examples in positive characteristic and cases where the generic stabilizer is finite are (mysteriously) linked to interesting geometric and number theoretic phenomena.

Finally, the program will include some of the exciting current research on groups acting on a product of Riemann surfaces, and the associated Beauville surfaces. The study of triangle groups also fits into this context. The recent work of Larsen, Marion and Lubotzky using deformation theory essentially proves Marion's conjecture on triangle generation of low-rank groups of Lie type and led to the new notion of ``saturation'' by finite quotients of a given type.

Conclusion: The finite simple groups find innovative applications across all four themes; recent such applications range from first order relational structures with rich automorphism groups, to edge-transitive embeddings of primitive graphs in Riemann surfaces, to the monodromy groups of such surfaces. The proposed workshop will focus on and highlight recent developments, and by bringing together researchers in these various fields, will promote interactions and further advances. Importantly, participants will have the opportunity not only to pursue their ongoing collaborations but also to establish connections with mathematicians working in related areas.

Finite permutation groups, graphs, geometries, and simple groups

Very recent results about finite permutation groups (many still unpublished) suggest that the finite simple groups still hold secrets about simply stated properties of finite permutation groups and the structures on which they act: for example, in most primitive permutation groups each element has a regular cycle (of maximum possible length, equal to the element order), and in most primitive groups each element has order less than a quarter of the number of points; and in both cases we can describe the exceptions. The young researchers involved in these results are among our list of participants.

New studies of symmetric structures demand strong new theoretical tools from permutation groups. The question of base size of permutation actions is of importance in computational group theory as well as in the study of the graph isomorphism problem. Recent research (by Burness, Fawcett, and more senior players) has thrown much light on the base sizes of actions of almost simple groups in particular -- to such an extent that the major obstacle to proving Pyber's base size conjecture is a sufficiently detailed understanding of finite imprimitive linear groups. Similarly Li's studies of edge-transitive embeddings of vertex-primitive graphs in compact Reimann surfaces leads to new factorisation problems for finite almost simple groups.

From the opposite point of view, breakthroughs in combinatorics and geometric group theory lend new methods for studying group actions: for instance, the exciting results of Helfgott on expansion in groups, extended recently by Pyber and Szab\'o, and independently by Breuillard, Green and Tao, have shown not only that Cayley graphs of bounded rank finite simple Lie type groups are expanders, but have also yielded a proof of the Weiss Conjecture of 1978 for locally primitive graphs involving only bounded rank composition factors. This highlights the classical Lie type groups of unbounded rank as those for which greater understanding is needed.

We believe that this workshop will spark substantial progress on some of these questions, and the workshop will provide a perfect opportunity to bring together senior researchers in this theme (such as Cameron, Guralnick, Li and Tiep) with young contributors in the area (Fawcett, Giudici, Guest, Morgan, Spiga, Verret). In addition these researchers work across the diferent foci of the workshop.

Infinite permutation groups and model theory

Over the last years there have also been major developments in infinite permutation group theory. One aspect here is the interaction between permutation group theory, combinatorics, model theory, and descriptive set theory, typically in the investigation of first order relational structures with rich automorphism groups. The connections between these fields are seen most clearly for permutation groups on countably infinite sets which are closed (in the topology of pointwise convergence) and oligomorphic (that is, have finitely many orbits on $k$-tuples for all $k$); these are exactly the automorphism groups of $\omega$-categorical structures, that is, first order structures determined up to isomorphism (among countable structures) by their first order theory.

Themes of current activity here include the following.

(a) The use of group theoretic means (O'Nan-Scott, Aschbacher's description of maximal subgroups of classical groups, representation theory) to obtain structural results for model-theoretically important classes (totally categorical structures, or much more generally, smoothly approximable structures, finite covers of well-understood structures). Recent progress by Macpherson, Kaplan and Simon towards showing that natural classes of automorphism groups are maximal closed subgroups of the symmetric group give rise to hope for a complete classification in certain settings.

(b) Infinite $2$-transitive groups are far from being classified, even in the split or the sharply $2$-transitive case. Here model theory comes in on the one hand as a means towards constructing examples as well as a setting of special cases like the assumption of finite Morley rank where such a classification would be essential. Recently the first non-split sharply 2-transitive groups were constructed by Rips, Segev and Tent. The existence of non-split sharply $2$-transitive groups is still open in all characteristics different from 2 and 3, but recent progress should be discussed to obtain hopefully a complete picture.

(c) We plan to further investigate properties which the full symmetric group $S$ on a countable set shares with various other closed oligomorphic groups like uncountable cofinality, the existence of conjugacy classes that are dense or even comeager in the automorphism group, or the Bergman property. Model theoretic methods have recently been used to obtain a general criterion for the abstract simplicity of such automorphism groups or in other cases to determine the full normal subgroup structure. The criterion applies to diverse examples such as the Urysohn space, random graphs, random hypergraphs, or random tournaments.

Totally disconnected locally compact groups and permutation groups

Infinite permutation groups with finite sub-degrees provide an approach to studying totally disconnected, locally compact groups (t.d.l.c groups). On one hand, every such group has associated infinite graphs, known as the `rough Cayley', `Schreier' or `Abels-Cayley' graphs, on which it acts transitively with finite sub-degrees while, on the other, every transitive group action on an infinite set having finite sub-degrees gives rise to a totally disconnected, locally compact group. New concepts such as the scale, tidy subgroups and flatness have been introduced in the structure theory of totally disconnected, locally compact groups and a typology for simple groups is being developed. This progress is leading to an exchange of ideas and problems with the theory of infinite permutation groups.

Primitivity of infinite permutation groups, as it applies to totally disconnected groups, has been studied in pioneering work by S. Smith on orbital digraphs and subdegree growth. Finiteness of sub-degrees in permutation representations corresponds to the structural fact that t.d.l.c. groups contain open profinite subgroups and the values of these sub-degrees correlate with values of the scale. Work on commensurators of profinite groups by Barnea, Ershov and Weigel initiated a study of how their profinite local structure influences the global structure of t.d.l.c. groups, which is being taken further by Caprace, Reid and Willis in work that bases a typology of simple t.d.l.c. groups on this local structure.

When the infinite set being permuted comprises the vertices, or other objects, in a geometry, the geometrical intuition resulting is a powerful method for understanding the group. For example, the flat-rank of Kac-Moody t.d.l.c. groups corresponds to the dimension of flats in their associated building. Automorphism groups of trees and buildings are, through the work of Tits, Haglund and Paulin, a rich source of t.d.l.c. groups. This work has inspired attempts at general constructions of canonical geometries for t.d.l.c. groups by Baumgartner, Ramagge and Willis, with one outcome being a `space of directions' in which it is possible to move to infinity within the group.

A link between actions on graphs and the scale and tidy subgroups was made early on by M\"oller when he characterised tidiness of a subgroup in terms of a certain graph being a regular rooted tree whose valency is the scale. A higher rank version of this, that characterises flatness in terms of the structure of a graph generalising the one considered by M\"oller, would give another translation between group structure and actions on graphs.

Algebraic groups, subgroup structure and permutation group actions The large majority of the finite quasisimple groups are finite groups of Lie type, which are obtained as fixed point subgroups of certain endomorphisms of reductive algebraic groups. A main aim is to understand the subgroup lattice of these groups, knowledge of which is required for the study of permutation actions of the groups. Well-known reduction theorems motivate the study of so-called irreducible triples $(X,Y,V)$, where $X$ is a subgroup of $Y$ and $V$ is an absolutely irreducible module for $Y$ on which $X$ acts irreducibly. Recent progress on the classification of such triples has been achieved in the work of Husen, Hiss and Magaard on imprimitive actions for finite groups, Burness, Ghandour, Marion and Testerman, on irreducible triples for classical algebraic groups. In addition, many people are actively contributing to the study of irreducible triples in the finite groups. In a different direction, the recent work of Litterick, Stewart, and Thomas aims at understanding the subgroup lattice of an exceptional type algebraic group $G$. They consider $G$-irreducible subgroups, and $G$-non completely reducible subgroups, as well as so-called Lie-primitive finite subgroups, work leading to whole new insights on the subgroup stucture of the exceptional groups.

In another direction, Burness, Guralnick and Saxl have undertaken an extensive study of the problem of base size for algebraic group actions, obtaining the precise value for almost all primitive actions of simple algebraic groups, and applying their results to the study of the essential dimension of simple algebraic groups. A related question, studied in on-going work of Guralnick, Lawther, Liebeck, and Martin, is that of regular orbits in algebraic group actions. They determine for which irreducible modules $V$ for a simple algebraic group, the generic stabilizer is non trivial. That is, when does there exist a non-empty open subset of $V$ such that for all points in the set the stabilizer is non trivial. There are new and interesting examples in positive characteristic and cases where the generic stabilizer is finite are (mysteriously) linked to interesting geometric and number theoretic phenomena.

Finally, the program will include some of the exciting current research on groups acting on a product of Riemann surfaces, and the associated Beauville surfaces. The study of triangle groups also fits into this context. The recent work of Larsen, Marion and Lubotzky using deformation theory essentially proves Marion's conjecture on triangle generation of low-rank groups of Lie type and led to the new notion of ``saturation'' by finite quotients of a given type.

Conclusion: The finite simple groups find innovative applications across all four themes; recent such applications range from first order relational structures with rich automorphism groups, to edge-transitive embeddings of primitive graphs in Riemann surfaces, to the monodromy groups of such surfaces. The proposed workshop will focus on and highlight recent developments, and by bringing together researchers in these various fields, will promote interactions and further advances. Importantly, participants will have the opportunity not only to pursue their ongoing collaborations but also to establish connections with mathematicians working in related areas.