About me

Geometric Group Theory

My MSc thesis was in geometric group theory. Together with my advisor we introduced disjoint sum norm and disjoint sum metric for any group \(G\). We related the boundedness of this metric to boundedness of Burnside’s groups in any bi-invariant word metric.

Construction of metric. Recall that for general two groups \(G, H\) their wreath product is defined to be the semidirect product

where \(\phi: G \rightarrow \text{Aut}(\bigoplus_G H)\) is given by translation action i.e. \(\phi(g)(\sigma)( \overline{g}) := \sigma(g^{-1}\overline{g})\). We simply write \(g \cdot \sigma\) instead of \(\phi(g)(\sigma)\). In our case \(H\) is always specialized to be \(\mathbb{Z}_2\) and we write \(\sigma = \delta_{g_1} + \dots + \delta_{g_k}\).

Suppose \(G\) is generated by set \(S\) and consider \(G\) embedded in \(\mathbb{Z}_2 \wr G\) by \(g \mapsto (0,g)\). Put \(S_1 := S \cup \{(\delta_e,e)\}\), where \(\delta_e\) is the Dirac’s delta on identity element in \(G\). Straighforward argument shows that \(\langle S_1 \rangle = \mathbb{Z_2} \wr G\). Now, let us consider

Careful observation on how elements \(\overline{s} \in \overline{S_1}\) look like motivated us to consider subset

Note that since \(\delta_e + (\delta_g + g^{-1} \cdot \delta_g) = \delta_g\), it follows \(T\) generates \(\bigoplus_G \mathbb{Z}_2\).

Definition. Let \(G\) be a group and \(T\) a generating subset of \(\bigoplus_G \mathbb{Z}_2\) as described above. The disjoint sum norm is defined to be

Below I present our main result.

Recall that Burnside group on \(n\) generators and exponent \(d\) is defined to be:

These groups are utterly mysterious and interesting for mathematicians, we know not to much about them. It is even hard to predict whether such a group is non-trivial, since we’re basically killing each word of length \(d\). Nevertheless, there are interesting combinatorial results showing a bunch of such a group are infinite!

Theorem 2. For any bi-invariant word metric \(|-|\) on the Burnside group \(B(n+1,2d)\) there is inequality

Unfortunately we were not able to compute the left-hand side of this inequality in terms of parameters. But it is interesting we found one metric which is dominated by every bi-invariant word metric. Of course it may happen that this result trivializes (that is why we didn’t publish it).

Algebraic Geometry

During my PhD studies (which I interrupted and switched into industry) I studied equivariant cohomology classes of singular algebraic varieties. I’ll just shortly give a glimpse of definitions and fundamental localization theorem. I take these notes below from the talk of my advisor: https://www.mimuw.edu.pl/~aweber/ps/hftalk2.pdf

Definition. Suppose topological group \(G\) acts on topological space \(X\) and let \(\Lambda\) be ring (of coefficients). The equivariant cohomology ring of this action is defined to be:

Of course the functor \(H^*(-;-)\) is an ordinary cohomology functor which I’ll not define here.

Remark 1. If \(G\) is trivial group, this is ordinary cohomology ring.

Remark 2. If \(X\) is contractible, it reduces to the ring of classifying space \(BG\).

Remark 3. If \(G\) acts freely on \(X\), then canonical map \(EG \times_G X \rightarrow X/G\) is a homotopy equivalence, and in consequence

Assume that \(T = (\mathbb{C}^*)^n\) acts on a compact space \(M\). Then, equivariant cohomology \(H^*_T(M)\) is a module over equivariant cohomology ring of the point

There is important

Theorem [Borel]. The restriction to the fixed set \(M^T\) of the action:

becomes isomorphism after localization in the multiplicative set generated by the nontrivial characters

The case when this fixed points set is finite is of particular interest, since it allows us to quite easily glue local cohomological information to the whole information. To sketch what I mean by that, consider push-forward functor

where \(p: M \rightarrow pt\) is collapse map.

Theorem [Berline-Vergne]. For \(\alpha \in H^ *_T(M)\) the integral can by computed by summation:

where of course \(e_F\) is Euler class of \(F\). Now, if \(M^T=\{p_0,p_1,\dots,p_n\} \), then the Euler class is product of weights

provided that \(T_pM=\bigoplus_{\lambda \in \Lambda} V_\lambda\). The integral along \(M\) is equal to the sum of fractions:

Here, I should explain some implicit identifications. For a torus action \(G=T\), we identify \(H^*_T(pt)\) with polynomial algebra spanned by characters of \(T\):

A character \(\lambda: T \rightarrow \mathbb{C}^*\) corresponds to an element of \(H^2_T(pt)\).

Exercise. Apply Berline-Vergne theorem to prove:

Here is my joint work together with my PhD advisor, published in Journal of Algebra: https://arxiv.org/pdf/2108.03598. We applied localization theorem in to compute interesting equivariant classes of Borel orbits of square-zero upper triangular matrices.