Colin D. Reid

I am currently a postdoc at ENS de Lyon, France.

My research focus is group theory, especially totally disconnected locally compact groups and group actions on infinite discrete structures.

I completed my PhD in 2010 at Queen Mary, University of London. Since then I have been employed in various research positions in Aachen (Germany), Göttingen (Germany), Louvain-la-Neuve (Belgium) and Newcastle (Australia).

What are totally disconnected locally compact (tdlc) groups?

Tdlc groups are a class of topological groups that occur as the symmetries of infinite discrete structures such as combinatorial graphs, partially ordered sets, cell complexes, algebraic structures such as fields, and so on, where the structure is typically “locally finite” in some sense (in order to ensure its symmetries are locally compact), and the nondiscreteness of the group reflects a “lack of rigidity” in the structure (in other words, one has nontrivial symmetries fixing many points). Tdlc groups also include the non-Archimedean counterpart to Lie groups. There are a few equivalent perspectives on what it means for a group to be tdlc:

A locally compact group is a group equipped with a locally compact Hausdorff topology, such that the group operations are continuous. In any such group, the connected component of the identity is a closed normal subgroup, and the quotient is a tdlc group. Connected locally compact groups are known to be pro-Lie groups, and one can form tdlc groups in a similar way over non-Archimedean fields, but in general, tdlc groups cannot be approximated by linear groups.
With the standard topology, the symmetric group of an infinite set is not locally compact, however it has many tdlc subgroups, namely those closed subgroups such that if you fix a large enough finite set of points, you have finite orbits on the remaining points.
A compact totally disconnected group is an inverse limit of finite groups, so it is residually finite in a strong sense. Every t.d.l.c. group G has a compact open subgroup U, and any two choices for U differ only by finite index, so U is commensurated in G (that is, it differs from its G-conjugates only by finite index). Given an abstract group G with a commensurated subgroup U, there are natural ‘completions’ of (G,U) to produce a tdlc group with a specified compact open subgroup; the original pair can be represented faithfully as long as U is residually finite. So tdlc group theory can be used to study commensurated subgroups in any group, analogous to using the profinite completion to study residually finite groups.

Research publications

Multiple transitivity except for a system of imprimitivity (2022), to appear in J. Group Theory, accepted manuscript
Growing trees from compact subgroups (with P.-E. Caprace and T. Marquis) 2022, to appear in Groups Geom Dyn., arXiv:2111.07066
Totally disconnected locally compact groups with just infinite locally normal subgroups, Israel J. Math. (2023), published online 3 June 2023, arXiv:2107.05329
Decomposition of locally compact coset spaces, J. London Math. Soc. 107 (2023), no. 1, 407–440, arXiv:2106.15180
Locally normal subgroups and ends of locally compact Kac–Moody groups (with P.-E. Caprace and T. Marquis), Münster J. Math. 15 (2022), no. 2, arXiv:2112.04760
Orientation of piecewise powers of a minimal homeomorphism, J. Australian Math. Soc. 113 (2022), no. 2, 226–256, arXiv:1812.00480
Chief factors in Polish groups (with P. R. Wesolek, with an appendix by F. Le Maître), Math. Proc. Camb. Phil. Soc. 173 (2022), no. 2, 239–296, arXiv:1509.00719
Discrete locally finite full groups of Cantor set homeomorphisms (with A. Garrido), Bull. London Math. Soc. 53 (2021), no. 4, 1228–1248, arXiv:2005.08167
A classification of the abelian minimal closed normal subgroups of locally compact second-countable groups, J. Group Theory 24 (2021), no. 3, 509–531, arXiv:2006.03925
Approximating simple locally compact groups by their dense locally compact subgroups (with P.-E. Caprace and P. R. Wesolek), Int. Math. Res. Not. (2021), no. 7, 5037–5110, arXiv:1706.07317
Topologically simple, totally disconnected, locally compact infinite matrix groups (with P. Groenhout and G. A. Willis,), J. Lie Theory 30 (2020), no. 4, 965–980, arXiv:1911.09956
Equicontinuity, orbit closures and invariant compact open sets for group actions on zero-dimensional spaces, Groups Geom. Dyn. 14 (2020), no. 2, 413–425, arXiv:1710.00627
Distal actions on coset spaces in totally disconnected, locally compact groups, J. Topology and Analysis 12 (2020), no. 2, 491–532, arXiv:1610.06696
On the residual and profinite closures of commensurated subgroups (with P.-E. Caprace, P. H. Kropholler and P. R. Wesolek), Math. Proc. Camb. Phil. Soc. 169 (2020), no. 2, 411–432, arXiv:1706.06853
Homomorphisms into totally disconnected, locally compact groups with dense image (with P. R. Wesolek), Forum Mathematicum 31 (2019), no. 3, 685–701, arXiv:1509.00156
Dense normal subgroups and chief factors in locally compact groups (with P. R. Wesolek), Proc. LMS 116 (2018), no. 4, 760–812, arXiv:1601.07317
The essentially chief series of a compactly generated locally compact group (with P. R. Wesolek), Math. Ann. 370 (2018), no. 1–2, 841–861, arXiv:1509.06593
Locally normal subgroups of totally disconnected groups. Part I: General theory (with P.-E. Caprace and G. A. Willis), Forum Math. Sigma 5 (2017), e11, 76pp, arXiv:1304.5144
Locally normal subgroups of totally disconnected groups. Part II: Compactly generated simple groups (with P.-E. Caprace and G. A. Willis), Forum Math. Sigma 5 (2017), e12, 89pp, arXiv:1401.3142
Multiply Heaving Bodies in the Time-Domain, Symmetry and Complex Resonances (with H. A. Wolgamot and M. H. Meylan), J. Fluids and Structures 69 (2017), 232–251.
Dynamics of flat actions on totally disconnected, locally compact groups, New York J. Math. 22 (2016), 115–190, arXiv:1503.01863
The number of profinite groups with a specified Sylow subgroup, J. Australian Math. Soc. 99 (2015), no. 1, 108–127, arXiv:1301.2112
Endomorphisms of profinite groups, Groups Geom. Dyn. 8 (2014), no. 2, 553–564, arXiv:1112.3916
Limits of contraction groups and the Tits core (with P.-E. Caprace and G. A. Willis) , J. Lie Theory 24 (2014), 957–967, arXiv:1304.6246
Local Sylow theory of totally disconnected, locally compact groups, J. Group Theory 16 (2013), no. 4, 535–555, arXiv:1111.7256
The Generalised Pro-Fitting Subgroup of a Profinite Group, Comm. Algebra vol 41 (2013), no. 1, 294–308, arXiv:0904.0424
Inverse system characterizations of the (hereditarily) just infinite property in profinite groups, Bull. LMS vol 44 (2012), no. 3, 413–425. Corrected version: arXiv:1708.08301
On finite groups whose Sylow subgroups have a bounded number of generators, Arch. Math 96 (2011), no. 3, 207–214, arXiv:1003.4722
Subgroups of finite index and the just infinite property, J. Algebra 324 (2010), 2219–2222, arXiv:0905.1624
On the structure of just infinite profinite groups, J. Algebra 324 (2010), 2249–2261, arXiv:0906.1771

Research preprints

Rigid stabilizers and local prosolubility for boundary-transitive actions on trees (2023), arXiv:2301.09078
A class of well-founded totally disconnected locally compact groups (2021), arXiv:2108.02952
Groups acting on trees with Tits’ independence property (P) (with S. M. Smith, with an appendix by S. Tornier), (2020), arXiv:2002.11766

Expository articles

An introduction to the local-to-global behaviour of groups acting on trees and the theory of local action diagrams (with S. M. Smith), arXiv:2309.05065
Normal subgroup structure of totally disconnected locally compact groups, in: 2016 MATRIX Annals, Springer, 2018.
Simon Smith’s construction of an uncountable family of simple totally disconnected, locally compact groups (with G. A. Willis), in: New Directions in Locally Compact Groups, CUP, 2018.
Locally normal subgroups of simple locally compact groups, (with P.-E. Caprace and G. A. Willis) C. R. Acad. Sci. Paris, Ser. I, 351 Nr. 17–18 (2013), 657–661, arXiv:1303.6755