| NSF Org: |
DMS Division Of Mathematical Sciences |
| Recipient: |
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| Initial Amendment Date: | July 6, 2012 |
| Latest Amendment Date: | July 6, 2012 |
| Award Number: | 1207868 |
| Award Instrument: | Standard Grant |
| Program Manager: |
Joanna Kania-Bartoszynska
jkaniaba@nsf.gov (703)292-4881 DMS Division Of Mathematical Sciences MPS Directorate for Mathematical and Physical Sciences |
| Start Date: | August 15, 2012 |
| End Date: | December 31, 2016 (Estimated) |
| Total Intended Award Amount: | $118,700.00 |
| Total Awarded Amount to Date: | $118,700.00 |
| Funds Obligated to Date: |
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| History of Investigator: |
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| Recipient Sponsored Research Office: |
202 HIMES HALL BATON ROUGE LA US 70803-0001 (225)578-2760 |
| Sponsor Congressional District: |
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| Primary Place of Performance: |
LA US 70803-4918 |
| Primary Place of
Performance Congressional District: |
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| Unique Entity Identifier (UEI): |
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| Parent UEI: |
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| NSF Program(s): |
TOPOLOGY, EPSCoR Co-Funding |
| Primary Program Source: |
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| Program Reference Code(s): |
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| Program Element Code(s): |
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| Award Agency Code: | 4900 |
| Fund Agency Code: | 4900 |
| Assistance Listing Number(s): | 47.049 |
ABSTRACT
A central theme in geometric group theory is to classify finitely presented groups up to quasi-isomtery. This gives rise to the study of properties of groups which are invariant up to quasi-isometry. The primary focus of this project is to study various quasi-isometry invariants which come from ``fillings'' in geometric spaces modeling the group. These include Dehn functions, which capture the difficulty of filling loops with disks or other surfaces, and divergence functions, which measure the rate at which geodesics emerging from a point move away from each other. One aspect of the project is to study these invariants and their higher dimensional analogs in a variety of classes of groups, such as non-positively curved groups, Coxeter groups and mapping class groups. Another is to understand the relationships between different filling functions in a given group.
Groups are fundamental objects of study in mathematics. A good example of a group is the collection of symmetries of an object. The result of composing symmetries (i.e. performing them one after another) is again a symmetry. Many groups can be specified by giving a finite presentation: a finite list of generators, and relations which indicate when two orders of composing elements give the same answer. Naturally, it is important to understand when two presentations define the same, or similar groups, but this is not always evident by examining the presentations. One of the themes in geometric group theory is to use geometric tools to address the question of determining whether the groups given by two presentations have the same large-scale features. The goal of this project is to conduct a detailed study of a class of such tools called filling functions. These investigations give us better insight into the structure of finitely presented groups.
PUBLICATIONS PRODUCED AS A RESULT OF THIS RESEARCH
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PROJECT OUTCOMES REPORT
Disclaimer
This Project Outcomes Report for the General Public is displayed verbatim as submitted by the Principal Investigator (PI) for this award. Any opinions, findings, and conclusions or recommendations expressed in this Report are those of the PI and do not necessarily reflect the views of the National Science Foundation; NSF has not approved or endorsed its content.
The principal goal of this project was to conduct research in geometric group theory. This is a vibrant field in mathematics that combines tools from topology and geometry to answer questions about groups. Groups are mathematical entities that arise naturally, for instance as collections of symmetries of objects.
As in many fields, a central question in in mathematics is to classify the objects of study. In geometric group theory, one seeks to classify groups in terms of their "large-scale geometry", that is, one seeks to understand exactly when two groups are "quasi-isometric" or "commensurable" to each other.
This project revolves around studying certain quasi-isometry invariants called filling invariants, and addressing the classification question for a class of groups called right-angled Coxeter groups. The PI, together with coauthors Thomas and Stark have made considerable progress towards understaning this classification. The results of the project have been written up as three papers:
1. P. Dani and A. Thomas, Divergence in right-angled Coxeter groups, Transactions of the American Mathematical Society, 367 (2015), no. 5.
2. P. Dani and A. Thomas, Bowditch's JSJ tree and the quasi-isometry classification of certain Coxeter groups, submitted.
3. P. Dani, E. Stark, and A. Thomas, Commensurability for certain right-angled Coxeter groups and geometric amalgams of free groups, submitted.
These have also beed posted on the preprint server Arxiv, and the latter two have been submitted to high quality journals. Further work has been conducted on certain filling invariants called Dehn functions by the PI's graduate student, which will be published as a dissertation in the future. The research in this project has inspired further research by other mathematicians.
Last Modified: 03/22/2017
Modified by: Pallavi Dani
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