 Thanks, and I wanted to thank the organizers again for giving me a chance to talk and for organizing this conference So I'm just going to do a brief outline first of what I'm going to be talking about today So first I'm going to talk a bit of just give a definition of what a Toro relatively hyperbolic group is and give some talk about some properties of them Next I'm going to talk about splitings of groups for those who are unfamiliar with this these ideas But I'm but I hope that it should seem pretty familiar to everyone Then I'm going to talk about certain couple of important types of splitings which are show up in my work And that are necessary to define what a with the structure of a floor in a tower is and then I'm going to talk about how we find floors using pre retractions and how we find pre retractions using first order logic. I'm not a logician. So I so I have like a little short introduction to some first order logic for this talk and then I'm going to talk about what I recently proved. So a group G is a finally is so I'm going to say is let X be the The Kaley graph of a finally generated group and let P be a collection of finally generated subgroups. We construct the cone to off Kaley graph X tilde by joining a unique cone point for each distinct left coset of an element of P and each vertex of that coset and X. And we make both X and X tilde into metric spaces by giving all like edges like one Now we call that a geodesic metric spaces delta hyperbolic if there exists a delta greater than or equal to zero so that for any geodesic triangle One side of the geodesic triangle or any side of the geodesic triangle is within a delta neighborhood of the other two sides. So this is the thin or slim triangles. I don't know which one is there's a couple different ways to call it and they mean different things, but they really mean the same thing. But ideas, those triangles look skinny in these sorts of spaces. Finally, generate group G is a hyperbolic group if the Kaley graph is a delta hyperbolic metric space. And similarly, a finally generated group is hyperbolic relative to a collection of finally generated subgroups if the case if the corresponding cone off Kaley graph is a delta hyperbolic space and we have this other condition that each edge. For each edge and for each end are only finitely many embedded loops of that length and containing E is sometimes known as the bounded coset penetration property. What What this does is one thing that you might know that you might recall if you've done sort of any metric any hyperbolic metric geometry before is that That higher rank abelian groups like rank two and above are not higher rank free abelian groups that is are not hyperbolic. There's that you can't get thin triangles in there. But the idea is that if you put If you put a cone point above these above the above these cosets of these very flat looking parts of your Kaley graph. It makes those the distances within those cosets smaller because the idea is that you can that between any two points inside there you can go up to the cone point and then go down so So they're inside that coset the maximal distance between any two points is to so this makes all of our sort of big flat subgroups. Small and so we could think about hyperbolicity relative to that so a total relatively hyperbolic group is a torsion free group. Which is hyperbolic relative to the candidacy representatives of its maximal non sick like abelian subgroups. So some I think the thing to state right here is is why you might have seen something like this before these are these things arise in the wild in mathematics as the fundamental groups of Hyperbolic manifolds with court torus cusp cross section. So the idea is that this is a manifold which is hyperbolic but it has a boundary component that looks like a torus right torsion free hyperbolic groups are our TRH relative to the trivial subgroup TRH groups are CSA or conjugate separated abelian, which means that if you have a maximal abelian subgroups it intersects its conjugates trivially And TRH groups have JSJ splitting which I use in my research. I'm going to talk a little bit about what those are after Briefly discussing what's sort of the idea of splittings and graphs of groups. So the first there are two basic splittings that all other splittings are based off of. The first one is the amalgamated free product. And the other one is the HNN extension. The amalgamated free product you are all familiar with. This is the van campens theorem thing right You where you can where if you take two spaces and attach them along a subspace. The fundamental group is an amalgamated free product of those two of the two individual groups over the Group of the subspace. The HNN extension is similar where it's sort of like where we take a subspace and we attach it to another copy of itself using a torus, which is why we get Get this extra generator for the fundamental group T because we're adding in a handle. So HNN extensions are like adding handles. Amalgamated free products are like gluing two spaces together. If a group can be expressed as an amalgamated free product or an HNN extension. This is known as a one edge splitting. And this is sort of This sort is a leading A leading terminology because more generally, splittings are defined are represented in terms of graphs of groups. So if you have a finite group and you have a and you have and you can assign and you assign a Group to each vertex in each edge so that the edge groups in inject into the into the end its endpoint groups. We can construct a graph of groups decomposition. So for example, if I have this one edge graph. You and V joined by E their fundamental defend the pie. One of this. This is the fundamental group of this graph of groups would be The U vertex group amalgamated with the GV vertex group over GE. And similarly, if we have a loop like this, we do an HNN extension and more generally for larger graphs. The pie one of lambda will be constructed inductively via by a series of amalgamated free products in HNN extensions. So given a group, we can actually recover how split, we can recover splittings of that group by studying how it acts on trees. And by studying these actions on trees. There's been a lot of Interesting work into sort of trying to find a canonical splitting for a group. And this led to the theory of J S J splittings. So for any finitely generated group G, there exists a J S J splitting, which is up to some equivalence maximal with respect to all splittings of G. And it's also invariant under the automorphism. J S J splittings. I like to think of them as like molecular diagrams for groups because they classify the vertex groups into a few classes and one of the most important classes among them are surface type vertex groups. What that means is that you have some vertex in our graph of groups. Such that it the corresponding vertex group for that vertex is pi one of a compact surface and there's a objective correspondence between the boundary curves and the edges coming out of that vertex so that the edge group there is equal to pi one of that curve that boundary curve. I'm going to have a picture of what this what one of these would look like on the next page. I just want to say that because I'm only dealing with things with negative or the characteristic the things of other characteristic negative one that don't have pseudo and also of the morphism are referred to later on as it being exceptional and other surfaces of negative or the characteristic are non exceptional. So this is what we're going to look at. So this is a picture of what a sick what a vertex type sorry a surface type vertex would look like. So this is the way you think about is that we have the fundamental group of a surface and we've just attached using h and using using amalgamated free products and h and n extensions this surface to some other space which is which is a kg one of what is a kg one one of this right. So we have some other space down here with the fundamental group of this lower vertex so the. In order to define what a tower is I need to define what a centered splitting is so a centered splitting has a single has a single well one surface type vertex and all the edges are z and they all join that that surface type vertex to some other vertex. Yeah, so all the edges come out of the. Now the interesting the reason why these the surface type works vertices are actually really interesting and give us some interesting things because it turns out that we can find often find ways to retract this this entire thing down onto the lower subgroups and I'll explain a little bit more about that in a moment. So, if we have a. If Q is the fundamental group of a compact surface and it's this it's a subgroup of G we call a we call P from Q to G a boundary preserving map with values in G. I'm going to explain the. Why I chose this later it's kind of a joke why I chose this notation here but so it we it's a boundary preserving map if it restricts the conjugation by an element of G on each boundary subgroup of the surface group. The map is non degenerate if the image is not a billion P is not an isomorphism onto one of the conjugates and the surface is non exceptional and we similarly can define the same sort of thing if we have a. If we have a surface type vertex in a splitting of some other subgroup G. Now, the reason why these are these boundary preserving maps are interesting is because they allow us to, like I said collapse down this, this space that we've constructed to represent our group onto the lower parts of it. So if lamb does a non exceptional centered splitting of a finally generated torsion free group a with central vertex subgroup Q and lower or exterior vertex subgroup B one, we can. They, there exists a non degenerate boundary preserving map from Q to a if and only if we're able to find these nice retractions as stated in in this proposition due to your Delby and screen us. So, this, this sort of idea of being able to retract your splitting leads into the definition of what a retractable splitting is so. I think the more important, important thing for our purposes is that a centered splitting is retractable if and only if it satisfies any of the equivalent conditions of the previous proposition. We can find that there's a boundary preserving map on the surf on the surface of our center center splitting that allows us to retract the group down onto the lower part onto the lower group. So now we can finally define a floor so floor is a. So G is a floor of us over subgroup H if either G is free product of eight of H was Z or if G has a retractable centered splitting with whose base, which is the. This is the base right here, this abstract free product of the conjugates of the of the bottom groups. If we have a retractable centered splitting with base isomorphic to H and a group is a tower if we can, if we can describe it as a series of floors right. So, how do we find floors and towers. Well, we actually find them with what are called pre retractions. So these pre retractions allow us to find the retractions we need in order to define the floor. Pre, so pre retractions. I mean, I don't have so much time and I want to be able to get to the first order logic stuff. Pre retractions basically they diff, they differ from the identity, their map that differs from the identity map by conjugating on work in a certain in a nice way on vertex and edge groups so we say the two morphisms from a to G are lambda related if we if they differ by an inner automorphism on all edge groups, if they differ by an inner automorphism on all non surface type or access exceptional surface type vertex groups and if it acts if it acts really as a boundary preserving all the exceptional on all the non exceptional vertex groups. And the reason why we want to find a pre retractions is because this allows us to find retractable splitting, which then allows us to find a floor and then a tower. So, if you have an abelian splitting of a group a which abelian splitting meaning that all the edge groups are abelian. And then, and every edge carries one abelian vertex and one non abelian vertex and the corresponding vast territory of the splitting is one a cylindrical near the vertices. If there exists a non injective pre retraction. With respect to lambda than that splitting is actually retractable in a. So here I got the idea for the, the, using that notation, because I when I thought about it was kind of like, I was imagining the pre retraction sort of breaking the top floors off your tower so you're just breaking these surface pieces off the top. And I got was I had come across this tarot card of the tower and I was working on this I thought that was a pretty funny coincidence. So I went with that. So now, now I need to talk a little bit about personal logic and the remaining two minutes I have. Basically, if you were interested in whether a group whether a, like a logical statement is is valid over a given group, right so for example, the statement G commutator of G and H is equal to one if and only if G and H can commute and G, and G, and the sentence for all x and y commutators equal to one is valid if and only if your group is abelian. So, similar to this related to this is the idea of an elementary theory. So it's the set of all sentence sentence is satisfied by your group. There was an important problem called Tarski's problem. Tarski first stated this in 1945 and it was proved in 2006 independently by Salah and Carl on the bitch me as a cough. He showed that all finitely generated non abelian free groups have the same first order theory. So I basically went on to show that being hyperbolic is, is a hit Salah and later Simone Andre went on to show that being hyperbolic is actually a first order invariant so the first order theory of a group is enough to detect the, the, sorry, the logic of a group. Yeah, the first order logic of a group is enough to detect hyperbolicity, which is a geometric thing, which is pretty crazy. An elementary embedding is says that when it's a bit stronger than being an elementary equivalence that is having the same elementary theory, it's that you have a sub sub group. So that for any first order formula, and any tuple of elements in H, it's that. The formula with the assignment of those that tuple is valid in H if and only if it's valid in G. So in particular, they have the same first order theory. In 2011 Chloe Perron proved that that if G is a torsion free hyperbolic group and H is elementarily embedding embedded then G has a structure of a hyperbolic tower over H. What's important is because what Salah did, which was the really hard part of what he did is they was the converse is that if you have, if G is a torsion free hyperbolic group and G nature torsion free hyperbolic groups and if G is a simple floor over H, then H is elementarily embedded in G, which is a really hard thing, and, and hasn't been proved in more general cases but what I worked on recently was was generalizing Perron's result to a total relatively hyperbolic groups. And just briefly, I'm not going to go into any details of how the proof how the proof work but the difficult part of it, what were the following. So that when you have a, if you could find that when you have a elementarily embedded subgroup you can find pre retractions. So, yeah, it's very hard to actually do and if I'd be happy to talk about the details later during the geometric group theory session. But that's it for now.