Andrew Sale

I have two principal strands of research. One involves Max Dehn's conjugacy problem, looking for ways to use the geometry of groups to understand more about the nature of conjugacy in infinite groups. In a different direction I have been investigating the properties of outer automorphism groups of graph products, such as right-angled Artin groups. I also have a couple of papers on wreath products, for example, with Ben Hayes, we showed that the wreath product of two sofic groups is sofic.

For more details, have a look at my research statement.

Scroll down, or click for: Preprints, Publications, Thesis, Talks.

Preprints:

• The outer automorphism group of a RACG is either large or virtually abelian, with Tim Susse
  arXiv.

Publications:

• Metric approximations of wreath products, with Ben Hayes
  arXiv, Annales de l'Institut Fourier.


• Vastness properties of automorphism groups of RAAGs, with Vincent Guirardel
  arXiv, Journal of Topology (2018).


• Permute and conjugate: the conjugacy problem in relatively hyperbolic groups, with Yago Antolín
  arXiv, MathSciNet, Bulletin of the LMS (2016).


• Conjugacy length in group extensions,
  arXiv, MathSciNet, Communications in Algebra (2016).


• The geometry of the conjugacy problem in lamplighter groups,
  arXiv, MathSciNet, Algebra and Computer Science, AMS Contemporary Math (2016).


• The geometry of the conjugacy problem in wreath products and free solvable groups,
  arXiv, MathSciNet, Journal of Group Theory (2015).


• Palindromic width of wreath products, metabelian groups, and max-n solvable groups, with Tim Riley
  arXiv, MathSciNet, Groups Complexity Cryptology (2015).


• Metric behaviour of the Magnus embedding,
  arXiv, MathSciNet, Geometriae Dedicata (2015).


• Bounded Conjugators For Real Hyperbolic and Unipotent Elements in Semisimple Lie Groups,
  arXiv, MathSciNet, Journal of Lie Theory (2014).

Thesis

My DPhil thesis, supervised by Cornelia Drutu, is available here.

The Length of Conjugators in Solvable Groups and Lattices of Semisimple Lie Groups, University of Oxford, Dec 2012.

Other papers:

• The wreath product of two sofic groups is sofic, with Ben Hayes
  arXiv.
  

  This paper was the initial version of what later became Metric approximations of wreath products. It is notably shorter at 8 pages, and much of the technical details required for generalising the result are not needed here. Any reader interested only in the result for sofic groups may find it easier to start here.



• Short conjugators in solvable groups,
  arXiv.
  

All results in this paper were either generalised in my sequel Conjugacy length in group extensions (2016), or cut and and put into The geometry of the conjugacy problem in lamplighter groups (2016).

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Invited Talks:

The outer automorphism group of a RACG is either large or virtually abelian:
• April 2018: Geometry of outer spaces and outer automorphism groups, University of Warwick, UK.
• Sept 2017: Cornell University.
• June 2017: 32nd Summer Conference on Topology and its Applications, University of Dayton.
• April 2017: Redbud Topology Conference, University of Arkansas.
Vastness properties of McCool groups of free groups and automorphisms of RAAGs:
• March 2017: University of Oklahoma, Norman, OK.
When all finite groups are involved in the outer automorphism group of a RAAG:
• January 2017: Non-positive curvature in action, at the Newton Institute, Cambridge, UK. (Video)
• August 2016: Vanderbilt University, Nashville TN.
• June 2016: Instituto de Ciencias Matemáticas, Madrid, Spain.
• June 2016: Bristol University, UK.
• March 2016: Cornell University, Ithaca NY.
The permutation conjugacy length function:
• April 2016: CUNY Graduate Center, New York, NY.
• January 2016: Vanderbilt University, Nashville TN.
Geometry of the conjugacy problem (or similar title):
• May 2015: Spring Topology and Dynamics Conference (Bowling Green, OH) (click here for the slides)
• March 2015: University of Wisconsin, Madison, WI.
• March 2015: University of Utah, Salt Lake City, UT.
Embedding solvable groups into Hilbert spaces:
• March 2014: Université de Rennes 1, France.
A Geometric Version of the Conjugacy Problem in Semisimple Lie Groups:
• January 2015: Joint Math Meeting, Special Session on Groups, Algorithms and Cryptography (San Antonio, TX) (click here for the slides)
• January 2014: Institut Fourier, Grenoble, France.
• September 2013: Université de Rennes 1, France.
• April 2013: Cornell University, Ithaca, NY.
• March 2013: Binghamton University, Binghamton, NY.
Finding Short Conjugators in Wreath Products and Free Solvable Groups (abstract):
• February 2013: Cornell University, Ithaca, NY.
• November 2012: University of Oxford, UK.
• May 2012: Université Paris-Sud 11, France.
• May 2012: ENS de Lyon, France.
• March 2012: North British Geometric Group Theory Meeting (a.k.a. GeNERators IV)